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Log-PDE Methods for Rough Signature Kernels (2404.02926v2)
Published 1 Apr 2024 in cs.LG and math.AP
Abstract: Signature kernels, inner products of path signatures, underpin several machine learning algorithms for multivariate time series analysis. For bounded variation paths, signature kernels were recently shown to solve a Goursat PDE. However, existing PDE solvers only use increments as input data, leading to first order approximation errors. These approaches become computationally intractable for highly oscillatory input paths, as they have to be resolved at a fine enough scale to accurately recover their signature kernel, resulting in significant time and memory complexities. In this paper, we extend the analysis to rough paths, and show, leveraging the framework of smooth rough paths, that the resulting rough signature kernels can be approximated by a novel system of PDEs whose coefficients involve higher order iterated integrals of the input rough paths. We show that this system of PDEs admits a unique solution and establish quantitative error bounds yielding a higher order approximation to rough signature kernels.
- H. Drucker, C.J. Burges, L. Kaufman, A. Smola, V. Vapnik, Support vector regression machines. Advances in neural information processing systems 9 (1996) [3] C. Brouard, M. Szafranski, F. d’Alché Buc, Input output kernel regression: Supervised and semi-supervised structured output prediction with operator-valued kernels. Journal of Machine Learning Research 17, np (2016) [4] B. Schölkopf, J.C. Platt, J. Shawe-Taylor, A.J. Smola, R.C. Williamson, Estimating the support of a high-dimensional distribution. Neural computation 13(7), 1443–1471 (2001) [5] A. Gretton, K.M. Borgwardt, M.J. Rasch, B. Schölkopf, A. Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Brouard, M. Szafranski, F. d’Alché Buc, Input output kernel regression: Supervised and semi-supervised structured output prediction with operator-valued kernels. Journal of Machine Learning Research 17, np (2016) [4] B. Schölkopf, J.C. Platt, J. Shawe-Taylor, A.J. Smola, R.C. Williamson, Estimating the support of a high-dimensional distribution. Neural computation 13(7), 1443–1471 (2001) [5] A. Gretton, K.M. Borgwardt, M.J. Rasch, B. Schölkopf, A. Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Schölkopf, J.C. Platt, J. Shawe-Taylor, A.J. Smola, R.C. Williamson, Estimating the support of a high-dimensional distribution. Neural computation 13(7), 1443–1471 (2001) [5] A. Gretton, K.M. Borgwardt, M.J. Rasch, B. Schölkopf, A. Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Gretton, K.M. Borgwardt, M.J. Rasch, B. Schölkopf, A. Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Advances in neural information processing systems 9 (1996) [3] C. Brouard, M. Szafranski, F. d’Alché Buc, Input output kernel regression: Supervised and semi-supervised structured output prediction with operator-valued kernels. Journal of Machine Learning Research 17, np (2016) [4] B. Schölkopf, J.C. Platt, J. Shawe-Taylor, A.J. Smola, R.C. Williamson, Estimating the support of a high-dimensional distribution. Neural computation 13(7), 1443–1471 (2001) [5] A. Gretton, K.M. Borgwardt, M.J. Rasch, B. Schölkopf, A. Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Brouard, M. Szafranski, F. d’Alché Buc, Input output kernel regression: Supervised and semi-supervised structured output prediction with operator-valued kernels. Journal of Machine Learning Research 17, np (2016) [4] B. Schölkopf, J.C. Platt, J. Shawe-Taylor, A.J. Smola, R.C. Williamson, Estimating the support of a high-dimensional distribution. Neural computation 13(7), 1443–1471 (2001) [5] A. Gretton, K.M. Borgwardt, M.J. Rasch, B. Schölkopf, A. Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Schölkopf, J.C. Platt, J. Shawe-Taylor, A.J. Smola, R.C. Williamson, Estimating the support of a high-dimensional distribution. Neural computation 13(7), 1443–1471 (2001) [5] A. Gretton, K.M. Borgwardt, M.J. Rasch, B. Schölkopf, A. Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Gretton, K.M. Borgwardt, M.J. Rasch, B. Schölkopf, A. Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). 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Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Gretton, K.M. Borgwardt, M.J. Rasch, B. Schölkopf, A. Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. 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Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Gretton, K.M. Borgwardt, M.J. Rasch, B. Schölkopf, A. Smola, A kernel two-sample test. The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- The Journal of Machine Learning Research 13(1), 723–773 (2012) [6] G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Wynne, A.B. Duncan, A kernel two-sample test for functional data. The Journal of Machine Learning Research 23(1), 3159–3209 (2022) [7] C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. 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Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. 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Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
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Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.K. Williams, C.E. Rasmussen, Gaussian processes for machine learning, vol. 2 (MIT press Cambridge, MA, 2006) [8] J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Lee, Y. Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as gaussian processes. arXiv preprint arXiv:1711.00165 (2017) [9] A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
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Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. 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Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. 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Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. 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Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Advances in neural information processing systems 31 (2018) [10] C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C.L. Li, W.C. Chang, Y. Cheng, Y. Yang, B. Póczos, Mmd gan: Towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584 (2017) [11] L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ L. Pacchiardi, Statistical inference in generative models using scoring rules. Ph.D. thesis, University of Oxford (2022) [12] T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). 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Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. 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Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Matsubara, J. Knoblauch, F.X. Briol, C.J. Oates, et al., Robust generalised bayesian inference for intractable likelihoods. Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Journal of the Royal Statistical Society Series B 84(3), 997–1022 (2022) [13] Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Chen, B. Hosseini, H. Owhadi, A.M. Stuart, Solving and learning nonlinear pdes with gaussian processes. Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Journal of Computational Physics 447, 110668 (2021) [14] P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Batlle, M. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning. Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Journal of Computational Physics 496, 112549 (2024) [15] T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- T.J. Lyons, Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998) [16] I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Chevyrev, A. Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- arXiv preprint arXiv:1603.03788 (2016) [17] I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ I. Perez Arribas, G.M. Goodwin, J.R. Geddes, T. Lyons, K.E. Saunders, A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder. Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Translational psychiatry 8(1), 274 (2018) [18] A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- A. Fermanian, Embedding and learning with signatures. Computational Statistics & Data Analysis 157, 107148 (2021) [19] T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cass, T. Lyons, X. Xu, Weighted signature kernels. The Annals of Applied Probability 34(1A), 585–626 (2024) [20] T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
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Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths. Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Ecole d’été de Probabilités de Saint-Flour XXXIV pp. 1–93 (2004) [21] F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ F.J. Király, H. Oberhauser, Kernels for sequentially ordered data. Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Journal of Machine Learning Research 20 (2019) [22] C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Bayesian learning from sequential data using gaussian processes with signature covariances, in International Conference on Machine Learning (PMLR, 2020), pp. 9548–9560 [23] M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- M. Lemercier, C. Salvi, T. Cass, E.V. Bonilla, T. Damoulas, T.J. Lyons, SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data, in International Conference on Machine Learning (PMLR, 2021), pp. 6233–6242 [24] M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- M. Lemercier, C. Salvi, T. Damoulas, E. Bonilla, T. Lyons, Distribution regression for sequential data, in International Conference on Artificial Intelligence and Statistics (PMLR, 2021), pp. 3754–3762 [25] C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- C. Salvi, M. Lemercier, C. Liu, B. Horvath, T. Damoulas, T. Lyons, Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Advances in Neural Information Processing Systems 34, 16635–16647 (2021) [26] T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Cochrane, P. Foster, V. Chhabra, M. Lemercier, T. Lyons, C. Salvi, Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel, in 2021 IEEE international conference on cyber security and resilience (CSR) (IEEE, 2021), pp. 35–40 [27] J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- J. Dyer, J. Fitzgerald, B. Rieck, S.M. Schmon, Approximate bayesian computation for panel data with signature maximum mean discrepancies, in NeurIPS 2022 Temporal Graph Learning Workshop (2022) [28] Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Z. Issa, B. Horvath, M. Lemercier, C. Salvi, Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
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McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Advances in Neural Information Processing Systems 36 (2024) [29] A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Fermanian, P. Marion, J.P. Vert, G. Biau, Framing rnn as a kernel method: A neural ode approach. Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Advances in Neural Information Processing Systems 34, 3121–3134 (2021) [30] N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ N. Muca Cirone, M. Lemercier, C. Salvi, Neural signature kernels as infinite-width-depth-limits of controlled resnets. arXiv e-prints pp. arXiv–2303 (2023) [31] A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. 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Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
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Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Pannier, C. Salvi, A path-dependent pde solver based on signature kernels. arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- arXiv preprint arXiv:2403.11738 (2024) [32] T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T.L. et al, Coropa computational rough paths (software library) (2010). URL http://coropa.sourceforge.net/ [33] J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Reizenstein, B. Graham, The iisignature library: efficient calculation of iterated-integral signatures and log signatures. arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). 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Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. 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Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- arXiv preprint arXiv:1802.08252 (2018) [34] P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P. Kidger, T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, in International Conference on Learning Representations (2020) [35] P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- P.R. Sam Morley, T. Lyons. Roughpy 0.1.0 - pypi (2023). URL https://pypi.org/project/RoughPy/ [36] C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Salvi, T. Cass, J. Foster, T. Lyons, W. Yang, The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- SIAM Journal on Mathematics of Data Science 3(3), 873–899 (2021) [37] A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A. Rahimi, B. Recht, Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007) [38] C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
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Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Toth, H. Oberhauser, Z. Szabo, Random fourier signature features. arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- arXiv preprint arXiv:2311.12214 (2023) [39] C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bellingeri, P.K. Friz, S. Paycha, R. Preiß, Smooth rough paths, their geometry and algebraic renormalization. Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Vietnam Journal of Mathematics 50(3), 719–761 (2022) [40] G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ G. Flint, T. Lyons, Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- arXiv preprint arXiv:1505.01298 (2015) [41] H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ H. Boedihardjo, X. Geng, N.P. Souris, Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Advances in Mathematics 364, 107043 (2020) [42] T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ T. Lyons, Rough paths, signatures and the modelling of functions on streams. Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Proceedings of the International Congress of Mathematicians, Korea (2014) [43] Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ Y. Boutaib, L.G. Gyurkó, T. Lyons, D. Yang, Dimension-free euler estimates of rough differential equations. Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Revue Roumaine de Mathmatiques Pures et Appliques, 59 (2014) [44] C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ C. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations (Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut …, 2023) [45] J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- J. Morrill, C. Salvi, P. Kidger, J. Foster, Neural rough differential equations for long time series, in International Conference on Machine Learning (PMLR, 2021), pp. 7829–7838 [46] B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- B. Walker, A.D. McLeod, T. Qin, Y. Cheng, H. Li, T. Lyons, Log neural controlled differential equations: The lie brackets make a difference. arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- arXiv preprint arXiv:2402.18512 (2024) [47] K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ K.T. Chen, Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- Annals of Mathematics 65(1), 163–178 (1957) [48] A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/ A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/
- A.T. et al. signax 0.2.1 - pypi (2024). URL https://pypi.org/project/RoughPy/