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Linearly-scalable learning of smooth low-dimensional patterns with permutation-aided entropic dimension reduction (2306.10287v1)
Published 17 Jun 2023 in cs.LG and math.OC
Abstract: In many data science applications, the objective is to extract appropriately-ordered smooth low-dimensional data patterns from high-dimensional data sets. This is challenging since common sorting algorithms are primarily aiming at finding monotonic orderings in low-dimensional data, whereas typical dimension reduction and feature extraction algorithms are not primarily designed for extracting smooth low-dimensional data patterns. We show that when selecting the Euclidean smoothness as a pattern quality criterium, both of these problems (finding the optimal 'crisp' data permutation and extracting the sparse set of permuted low-dimensional smooth patterns) can be efficiently solved numerically as one unsupervised entropy-regularized iterative optimization problem. We formulate and prove the conditions for monotonicity and convergence of this linearly-scalable (in dimension) numerical procedure, with the iteration cost scaling of $\mathcal{O}(DT2)$, where $T$ is the size of the data statistics and $D$ is a feature space dimension. The efficacy of the proposed method is demonstrated through the examination of synthetic examples as well as a real-world application involving the identification of smooth bankruptcy risk minimizing transition patterns from high-dimensional economical data. The results showcase that the statistical properties of the overall time complexity of the method exhibit linear scaling in the dimensionality $D$ within the specified confidence intervals.
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(ICM) 2010 (2010). http://publications.mi.fu-berlin.de/991/ Schütte and Sarich [2013] Schütte, C., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, (2013) Röblitz and Weber [2013] Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. [2017] Mena, G., Belanger, D., Munoz, G., Snoek, J.: Sinkhorn networks: Using optimal transport techniques to learn permutations. In: NIPS Workshop in Optimal Transport and Machine Learning, vol. 3 (2017) Mena et al. [2020] Mena, G., Varol, E., Nejatbakhsh, A., Yemini, E., Paninski, L.: Sinkhorn permutation variational marginal inference. In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. Proceedings of Machine Learning Research, vol. 118, pp. 1–9. Cambridge MA: JMLR, Cambridge, US (2020). https://proceedings.mlr.press/v118/mena20a.html Nikolentzos et al. [2023] Nikolentzos, G., Dasoulas, G., Vazirgiannis, M.: Permute me softly: Learning soft permutations for graph representations. IEEE Transactions on Pattern Analysis & Machine Intelligence 45(04), 5087–5098 (2023) https://doi.org/10.1109/TPAMI.2022.3188911 Artin [2013] Artin, M.: Algebra. Pearson Education, London, UK (2013) Calvetti et al. [2000] Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the l-curve for large discrete ill-posed problems. Journal of computational and applied mathematics 123(1-2), 423–446 (2000) Liang et al. [2016] Liang, D., Lu, C.-C., Tsai, C.-F., Shih, G.-A.: Financial ratios and corporate governance indicators in bankruptcy prediction: A comprehensive study. European Journal of Operational Research 252(2), 561–572 (2016) https://doi.org/10.1016/j.ejor.2016.01.012 Chen and Guestrin [2016] Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’16, pp. 785–794. ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785 . http://doi.acm.org/10.1145/2939672.2939785 LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Birkhoff, G.: Three observations on linear algebra. Universidad Nacional de Tucuman. Revista A. 5, 147–151 (1946) von Neumann [1953] Neumann, J.: In: Kuhn, H.W., Tucker, A.W. (eds.) 1. A Certain Zero-sum Two-person Game Equivalent to the Optimal Assignment Problem, pp. 5–12. Princeton University Press, Princeton (1953). https://doi.org/10.1515/9781400881970-002 . https://doi.org/10.1515/9781400881970-002 Ziegler [1995] Ziegler, G.M.: Lectures on Polytopes. Springer, New York (1995). http://www.worldcat.org/search?qt=worldcat_org_all&q=9780387943657 Gagniuc [2017] Gagniuc, P.: Markov Chains: From Theory to Implementation and Experimentation. John Wiley and Sons, Toronto, Canada (2017). https://doi.org/10.1002/9781119387596 Deuflhard et al. [2000] Deuflhard, P., Huisinga, W., Fischer, A., Schütte, C.: Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Linear Algebra and its Applications 315, 39–59 (2000) Conrad et al. [2010] Conrad, N.D., Sarich, M., Schütte, C.: On markov state models for metastable processes. In: Proceedings of the International Congress of Mathematics, Hyderabad, India, Section Invited Talks. (ICM) 2010 (2010). http://publications.mi.fu-berlin.de/991/ Schütte and Sarich [2013] Schütte, C., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, (2013) Röblitz and Weber [2013] Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. [2017] Mena, G., Belanger, D., Munoz, G., Snoek, J.: Sinkhorn networks: Using optimal transport techniques to learn permutations. In: NIPS Workshop in Optimal Transport and Machine Learning, vol. 3 (2017) Mena et al. [2020] Mena, G., Varol, E., Nejatbakhsh, A., Yemini, E., Paninski, L.: Sinkhorn permutation variational marginal inference. In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. Proceedings of Machine Learning Research, vol. 118, pp. 1–9. Cambridge MA: JMLR, Cambridge, US (2020). https://proceedings.mlr.press/v118/mena20a.html Nikolentzos et al. [2023] Nikolentzos, G., Dasoulas, G., Vazirgiannis, M.: Permute me softly: Learning soft permutations for graph representations. IEEE Transactions on Pattern Analysis & Machine Intelligence 45(04), 5087–5098 (2023) https://doi.org/10.1109/TPAMI.2022.3188911 Artin [2013] Artin, M.: Algebra. Pearson Education, London, UK (2013) Calvetti et al. [2000] Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the l-curve for large discrete ill-posed problems. Journal of computational and applied mathematics 123(1-2), 423–446 (2000) Liang et al. [2016] Liang, D., Lu, C.-C., Tsai, C.-F., Shih, G.-A.: Financial ratios and corporate governance indicators in bankruptcy prediction: A comprehensive study. European Journal of Operational Research 252(2), 561–572 (2016) https://doi.org/10.1016/j.ejor.2016.01.012 Chen and Guestrin [2016] Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’16, pp. 785–794. ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785 . http://doi.acm.org/10.1145/2939672.2939785 LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Neumann, J.: In: Kuhn, H.W., Tucker, A.W. (eds.) 1. A Certain Zero-sum Two-person Game Equivalent to the Optimal Assignment Problem, pp. 5–12. Princeton University Press, Princeton (1953). https://doi.org/10.1515/9781400881970-002 . https://doi.org/10.1515/9781400881970-002 Ziegler [1995] Ziegler, G.M.: Lectures on Polytopes. Springer, New York (1995). http://www.worldcat.org/search?qt=worldcat_org_all&q=9780387943657 Gagniuc [2017] Gagniuc, P.: Markov Chains: From Theory to Implementation and Experimentation. John Wiley and Sons, Toronto, Canada (2017). https://doi.org/10.1002/9781119387596 Deuflhard et al. [2000] Deuflhard, P., Huisinga, W., Fischer, A., Schütte, C.: Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Linear Algebra and its Applications 315, 39–59 (2000) Conrad et al. [2010] Conrad, N.D., Sarich, M., Schütte, C.: On markov state models for metastable processes. In: Proceedings of the International Congress of Mathematics, Hyderabad, India, Section Invited Talks. (ICM) 2010 (2010). http://publications.mi.fu-berlin.de/991/ Schütte and Sarich [2013] Schütte, C., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, (2013) Röblitz and Weber [2013] Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. [2017] Mena, G., Belanger, D., Munoz, G., Snoek, J.: Sinkhorn networks: Using optimal transport techniques to learn permutations. In: NIPS Workshop in Optimal Transport and Machine Learning, vol. 3 (2017) Mena et al. [2020] Mena, G., Varol, E., Nejatbakhsh, A., Yemini, E., Paninski, L.: Sinkhorn permutation variational marginal inference. In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. Proceedings of Machine Learning Research, vol. 118, pp. 1–9. Cambridge MA: JMLR, Cambridge, US (2020). https://proceedings.mlr.press/v118/mena20a.html Nikolentzos et al. [2023] Nikolentzos, G., Dasoulas, G., Vazirgiannis, M.: Permute me softly: Learning soft permutations for graph representations. IEEE Transactions on Pattern Analysis & Machine Intelligence 45(04), 5087–5098 (2023) https://doi.org/10.1109/TPAMI.2022.3188911 Artin [2013] Artin, M.: Algebra. Pearson Education, London, UK (2013) Calvetti et al. [2000] Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the l-curve for large discrete ill-posed problems. Journal of computational and applied mathematics 123(1-2), 423–446 (2000) Liang et al. [2016] Liang, D., Lu, C.-C., Tsai, C.-F., Shih, G.-A.: Financial ratios and corporate governance indicators in bankruptcy prediction: A comprehensive study. European Journal of Operational Research 252(2), 561–572 (2016) https://doi.org/10.1016/j.ejor.2016.01.012 Chen and Guestrin [2016] Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’16, pp. 785–794. ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785 . http://doi.acm.org/10.1145/2939672.2939785 LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Ziegler, G.M.: Lectures on Polytopes. Springer, New York (1995). http://www.worldcat.org/search?qt=worldcat_org_all&q=9780387943657 Gagniuc [2017] Gagniuc, P.: Markov Chains: From Theory to Implementation and Experimentation. John Wiley and Sons, Toronto, Canada (2017). https://doi.org/10.1002/9781119387596 Deuflhard et al. [2000] Deuflhard, P., Huisinga, W., Fischer, A., Schütte, C.: Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Linear Algebra and its Applications 315, 39–59 (2000) Conrad et al. [2010] Conrad, N.D., Sarich, M., Schütte, C.: On markov state models for metastable processes. In: Proceedings of the International Congress of Mathematics, Hyderabad, India, Section Invited Talks. (ICM) 2010 (2010). http://publications.mi.fu-berlin.de/991/ Schütte and Sarich [2013] Schütte, C., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, (2013) Röblitz and Weber [2013] Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. 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Linear Algebra and its Applications 315, 39–59 (2000) Conrad et al. [2010] Conrad, N.D., Sarich, M., Schütte, C.: On markov state models for metastable processes. In: Proceedings of the International Congress of Mathematics, Hyderabad, India, Section Invited Talks. (ICM) 2010 (2010). http://publications.mi.fu-berlin.de/991/ Schütte and Sarich [2013] Schütte, C., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, (2013) Röblitz and Weber [2013] Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. 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(ICM) 2010 (2010). http://publications.mi.fu-berlin.de/991/ Schütte and Sarich [2013] Schütte, C., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, (2013) Röblitz and Weber [2013] Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. 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[2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. [2017] Mena, G., Belanger, D., Munoz, G., Snoek, J.: Sinkhorn networks: Using optimal transport techniques to learn permutations. In: NIPS Workshop in Optimal Transport and Machine Learning, vol. 3 (2017) Mena et al. [2020] Mena, G., Varol, E., Nejatbakhsh, A., Yemini, E., Paninski, L.: Sinkhorn permutation variational marginal inference. In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. Proceedings of Machine Learning Research, vol. 118, pp. 1–9. Cambridge MA: JMLR, Cambridge, US (2020). https://proceedings.mlr.press/v118/mena20a.html Nikolentzos et al. 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In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’16, pp. 785–794. ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785 . http://doi.acm.org/10.1145/2939672.2939785 LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. 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ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785 . http://doi.acm.org/10.1145/2939672.2939785 LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. 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Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
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Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
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In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. Proceedings of Machine Learning Research, vol. 118, pp. 1–9. Cambridge MA: JMLR, Cambridge, US (2020). https://proceedings.mlr.press/v118/mena20a.html Nikolentzos et al. [2023] Nikolentzos, G., Dasoulas, G., Vazirgiannis, M.: Permute me softly: Learning soft permutations for graph representations. IEEE Transactions on Pattern Analysis & Machine Intelligence 45(04), 5087–5098 (2023) https://doi.org/10.1109/TPAMI.2022.3188911 Artin [2013] Artin, M.: Algebra. Pearson Education, London, UK (2013) Calvetti et al. [2000] Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the l-curve for large discrete ill-posed problems. Journal of computational and applied mathematics 123(1-2), 423–446 (2000) Liang et al. [2016] Liang, D., Lu, C.-C., Tsai, C.-F., Shih, G.-A.: Financial ratios and corporate governance indicators in bankruptcy prediction: A comprehensive study. European Journal of Operational Research 252(2), 561–572 (2016) https://doi.org/10.1016/j.ejor.2016.01.012 Chen and Guestrin [2016] Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’16, pp. 785–794. ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785 . http://doi.acm.org/10.1145/2939672.2939785 LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Deuflhard, P., Huisinga, W., Fischer, A., Schütte, C.: Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Linear Algebra and its Applications 315, 39–59 (2000) Conrad et al. [2010] Conrad, N.D., Sarich, M., Schütte, C.: On markov state models for metastable processes. In: Proceedings of the International Congress of Mathematics, Hyderabad, India, Section Invited Talks. (ICM) 2010 (2010). http://publications.mi.fu-berlin.de/991/ Schütte and Sarich [2013] Schütte, C., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, (2013) Röblitz and Weber [2013] Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. [2017] Mena, G., Belanger, D., Munoz, G., Snoek, J.: Sinkhorn networks: Using optimal transport techniques to learn permutations. In: NIPS Workshop in Optimal Transport and Machine Learning, vol. 3 (2017) Mena et al. [2020] Mena, G., Varol, E., Nejatbakhsh, A., Yemini, E., Paninski, L.: Sinkhorn permutation variational marginal inference. In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. Proceedings of Machine Learning Research, vol. 118, pp. 1–9. Cambridge MA: JMLR, Cambridge, US (2020). https://proceedings.mlr.press/v118/mena20a.html Nikolentzos et al. [2023] Nikolentzos, G., Dasoulas, G., Vazirgiannis, M.: Permute me softly: Learning soft permutations for graph representations. IEEE Transactions on Pattern Analysis & Machine Intelligence 45(04), 5087–5098 (2023) https://doi.org/10.1109/TPAMI.2022.3188911 Artin [2013] Artin, M.: Algebra. Pearson Education, London, UK (2013) Calvetti et al. [2000] Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the l-curve for large discrete ill-posed problems. Journal of computational and applied mathematics 123(1-2), 423–446 (2000) Liang et al. [2016] Liang, D., Lu, C.-C., Tsai, C.-F., Shih, G.-A.: Financial ratios and corporate governance indicators in bankruptcy prediction: A comprehensive study. European Journal of Operational Research 252(2), 561–572 (2016) https://doi.org/10.1016/j.ejor.2016.01.012 Chen and Guestrin [2016] Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’16, pp. 785–794. ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785 . http://doi.acm.org/10.1145/2939672.2939785 LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. 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Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Conrad, N.D., Sarich, M., Schütte, C.: On markov state models for metastable processes. In: Proceedings of the International Congress of Mathematics, Hyderabad, India, Section Invited Talks. (ICM) 2010 (2010). http://publications.mi.fu-berlin.de/991/ Schütte and Sarich [2013] Schütte, C., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, (2013) Röblitz and Weber [2013] Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. [2017] Mena, G., Belanger, D., Munoz, G., Snoek, J.: Sinkhorn networks: Using optimal transport techniques to learn permutations. In: NIPS Workshop in Optimal Transport and Machine Learning, vol. 3 (2017) Mena et al. [2020] Mena, G., Varol, E., Nejatbakhsh, A., Yemini, E., Paninski, L.: Sinkhorn permutation variational marginal inference. In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. Proceedings of Machine Learning Research, vol. 118, pp. 1–9. Cambridge MA: JMLR, Cambridge, US (2020). https://proceedings.mlr.press/v118/mena20a.html Nikolentzos et al. [2023] Nikolentzos, G., Dasoulas, G., Vazirgiannis, M.: Permute me softly: Learning soft permutations for graph representations. IEEE Transactions on Pattern Analysis & Machine Intelligence 45(04), 5087–5098 (2023) https://doi.org/10.1109/TPAMI.2022.3188911 Artin [2013] Artin, M.: Algebra. Pearson Education, London, UK (2013) Calvetti et al. [2000] Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the l-curve for large discrete ill-posed problems. Journal of computational and applied mathematics 123(1-2), 423–446 (2000) Liang et al. [2016] Liang, D., Lu, C.-C., Tsai, C.-F., Shih, G.-A.: Financial ratios and corporate governance indicators in bankruptcy prediction: A comprehensive study. European Journal of Operational Research 252(2), 561–572 (2016) https://doi.org/10.1016/j.ejor.2016.01.012 Chen and Guestrin [2016] Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’16, pp. 785–794. ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785 . http://doi.acm.org/10.1145/2939672.2939785 LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Schütte, C., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, (2013) Röblitz and Weber [2013] Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. [2017] Mena, G., Belanger, D., Munoz, G., Snoek, J.: Sinkhorn networks: Using optimal transport techniques to learn permutations. In: NIPS Workshop in Optimal Transport and Machine Learning, vol. 3 (2017) Mena et al. [2020] Mena, G., Varol, E., Nejatbakhsh, A., Yemini, E., Paninski, L.: Sinkhorn permutation variational marginal inference. In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. Proceedings of Machine Learning Research, vol. 118, pp. 1–9. Cambridge MA: JMLR, Cambridge, US (2020). https://proceedings.mlr.press/v118/mena20a.html Nikolentzos et al. [2023] Nikolentzos, G., Dasoulas, G., Vazirgiannis, M.: Permute me softly: Learning soft permutations for graph representations. IEEE Transactions on Pattern Analysis & Machine Intelligence 45(04), 5087–5098 (2023) https://doi.org/10.1109/TPAMI.2022.3188911 Artin [2013] Artin, M.: Algebra. Pearson Education, London, UK (2013) Calvetti et al. [2000] Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the l-curve for large discrete ill-posed problems. Journal of computational and applied mathematics 123(1-2), 423–446 (2000) Liang et al. [2016] Liang, D., Lu, C.-C., Tsai, C.-F., Shih, G.-A.: Financial ratios and corporate governance indicators in bankruptcy prediction: A comprehensive study. European Journal of Operational Research 252(2), 561–572 (2016) https://doi.org/10.1016/j.ejor.2016.01.012 Chen and Guestrin [2016] Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’16, pp. 785–794. ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785 . http://doi.acm.org/10.1145/2939672.2939785 LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. [2017] Mena, G., Belanger, D., Munoz, G., Snoek, J.: Sinkhorn networks: Using optimal transport techniques to learn permutations. In: NIPS Workshop in Optimal Transport and Machine Learning, vol. 3 (2017) Mena et al. [2020] Mena, G., Varol, E., Nejatbakhsh, A., Yemini, E., Paninski, L.: Sinkhorn permutation variational marginal inference. In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. Proceedings of Machine Learning Research, vol. 118, pp. 1–9. Cambridge MA: JMLR, Cambridge, US (2020). https://proceedings.mlr.press/v118/mena20a.html Nikolentzos et al. [2023] Nikolentzos, G., Dasoulas, G., Vazirgiannis, M.: Permute me softly: Learning soft permutations for graph representations. IEEE Transactions on Pattern Analysis & Machine Intelligence 45(04), 5087–5098 (2023) https://doi.org/10.1109/TPAMI.2022.3188911 Artin [2013] Artin, M.: Algebra. Pearson Education, London, UK (2013) Calvetti et al. [2000] Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the l-curve for large discrete ill-posed problems. Journal of computational and applied mathematics 123(1-2), 423–446 (2000) Liang et al. [2016] Liang, D., Lu, C.-C., Tsai, C.-F., Shih, G.-A.: Financial ratios and corporate governance indicators in bankruptcy prediction: A comprehensive study. European Journal of Operational Research 252(2), 561–572 (2016) https://doi.org/10.1016/j.ejor.2016.01.012 Chen and Guestrin [2016] Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’16, pp. 785–794. ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785 . http://doi.acm.org/10.1145/2939672.2939785 LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. [2017] Mena, G., Belanger, D., Munoz, G., Snoek, J.: Sinkhorn networks: Using optimal transport techniques to learn permutations. In: NIPS Workshop in Optimal Transport and Machine Learning, vol. 3 (2017) Mena et al. [2020] Mena, G., Varol, E., Nejatbakhsh, A., Yemini, E., Paninski, L.: Sinkhorn permutation variational marginal inference. In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. Proceedings of Machine Learning Research, vol. 118, pp. 1–9. 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Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
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(ICM) 2010 (2010). http://publications.mi.fu-berlin.de/991/ Schütte and Sarich [2013] Schütte, C., Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, (2013) Röblitz and Weber [2013] Röblitz, S., Weber, M.: Fuzzy spectral clustering by pcca+: application to markov state models and data classification. Advances in Data Analysis and Classification 7(2), 147–179 (2013) https://doi.org/10.1007/s11634-013-0134-6 Gerber and Horenko [2015] Gerber, S., Horenko, I.: Improving clustering by imposing network information. Science Advances 1(7), 1500163 (2015) https://doi.org/10.1126/sciadv.1500163 https://www.science.org/doi/pdf/10.1126/sciadv.1500163 Aflalo et al. [2015] Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences 112(10), 2942–2947 (2015) https://doi.org/10.1073/pnas.1401651112 https://www.pnas.org/doi/pdf/10.1073/pnas.1401651112 Mena et al. 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Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Mena, G., Belanger, D., Munoz, G., Snoek, J.: Sinkhorn networks: Using optimal transport techniques to learn permutations. In: NIPS Workshop in Optimal Transport and Machine Learning, vol. 3 (2017) Mena et al. [2020] Mena, G., Varol, E., Nejatbakhsh, A., Yemini, E., Paninski, L.: Sinkhorn permutation variational marginal inference. In: Zhang, C., Ruiz, F., Bui, T., Dieng, A.B., Liang, D. (eds.) Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference. 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Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
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Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
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Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
- LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015) Horenko [2020] Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
- Horenko, I.: On a scalable entropic breaching of the overfitting barrier for small data problems in machine learning. Neural Computation 32(8), 1563–1579 (2020) https://doi.org/10.1162/neco_a_01296 Vecchi et al. [2022] Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
- Vecchi, E., PospÃÅ¡il, L., Albrecht, S., O’Kane, T.J., Horenko, I.: eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Computation 34(5), 1220–1255 (2022) https://doi.org/10.1162/neco_a_01490 https://direct.mit.edu/neco/article-pdf/34/5/1220/2008663/neco_a_01490.pdf Horenko [2022] Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I., Vecchi, E., KardoÅ¡, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
- Horenko, I.: Cheap robust learning of data anomalies with analytically solvable entropic outlier sparsification. Proceedings of the National Academy of Sciences 119(9), 2119659119 (2022) https://doi.org/10.1073/pnas.2119659119 https://www.pnas.org/doi/pdf/10.1073/pnas.2119659119 Horenko et al. [2023] Horenko, I., Vecchi, E., Kardoš, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120 Horenko, I., Vecchi, E., Kardoš, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
- Horenko, I., Vecchi, E., Kardoš, J., Wächter, A., Schenk, O., O’Kane, T.J., Gagliardini, P., Gerber, S.: On cheap entropy-sparsified regression learning. Proceedings of the National Academy of Sciences 120(1), 2214972120 (2023) https://doi.org/10.1073/pnas.2214972120 https://www.pnas.org/doi/pdf/10.1073/pnas.2214972120
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