2000 character limit reached
Quantum Circuit $C^*$-algebra Net (2404.06218v1)
Published 9 Apr 2024 in cs.LG, math.OA, and quant-ph
Abstract: This paper introduces quantum circuit $C*$-algebra net, which provides a connection between $C*$-algebra nets proposed in classical machine learning and quantum circuits. Using $C*$-algebra, a generalization of the space of complex numbers, we can represent quantum gates as weight parameters of a neural network. By introducing additional parameters, we can induce interaction among multiple circuits constructed by quantum gates. This interaction enables the circuits to share information among them, which contributes to improved generalization performance in machine learning tasks. As an application, we propose to use the quantum circuit $C*$-algebra net to encode classical data into quantum states, which enables us to integrate classical data into quantum algorithms. Numerical results demonstrate that the interaction among circuits improves performance significantly in image classification, and encoded data by the quantum circuit $C*$-algebra net are useful for downstream quantum machine learning tasks.
- Lance, E.C.: Hilbert C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-modules – a Toolkit for Operator Algebraists. London Mathematical Society Lecture Note Series, vol. 210. Cambridge University Press, New York (1995) Bru and de Siqueira Pedra [2023] Bru, J.-B., Siqueira Pedra, W.A.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-Algebras and Mathematical Foundations of Quantum Statistical Mechanics. Springer, Cham (2023) Hashimoto et al. [2021] Hashimoto, Y., Ishikawa, I., Ikeda, M., Komura, F., Katsura, T., Kawahara, Y.: Reproducing kernel Hilbert C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-module and kernel mean embeddings. Journal of Machine Learning Research 22(267), 1–56 (2021) Hasimoto et al. [2023a] Hasimoto, Y., Ikeda, M., Kadri, H.: Learning in RKHM: a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic twist for kernel machines. In: Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS) (2023) Hasimoto et al. [2023b] Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bru, J.-B., Siqueira Pedra, W.A.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-Algebras and Mathematical Foundations of Quantum Statistical Mechanics. Springer, Cham (2023) Hashimoto et al. [2021] Hashimoto, Y., Ishikawa, I., Ikeda, M., Komura, F., Katsura, T., Kawahara, Y.: Reproducing kernel Hilbert C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-module and kernel mean embeddings. Journal of Machine Learning Research 22(267), 1–56 (2021) Hasimoto et al. [2023a] Hasimoto, Y., Ikeda, M., Kadri, H.: Learning in RKHM: a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic twist for kernel machines. In: Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS) (2023) Hasimoto et al. [2023b] Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Ishikawa, I., Ikeda, M., Komura, F., Katsura, T., Kawahara, Y.: Reproducing kernel Hilbert C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-module and kernel mean embeddings. Journal of Machine Learning Research 22(267), 1–56 (2021) Hasimoto et al. [2023a] Hasimoto, Y., Ikeda, M., Kadri, H.: Learning in RKHM: a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic twist for kernel machines. In: Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS) (2023) Hasimoto et al. [2023b] Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hasimoto, Y., Ikeda, M., Kadri, H.: Learning in RKHM: a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic twist for kernel machines. In: Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS) (2023) Hasimoto et al. [2023b] Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Bru, J.-B., Siqueira Pedra, W.A.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-Algebras and Mathematical Foundations of Quantum Statistical Mechanics. Springer, Cham (2023) Hashimoto et al. [2021] Hashimoto, Y., Ishikawa, I., Ikeda, M., Komura, F., Katsura, T., Kawahara, Y.: Reproducing kernel Hilbert C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-module and kernel mean embeddings. Journal of Machine Learning Research 22(267), 1–56 (2021) Hasimoto et al. [2023a] Hasimoto, Y., Ikeda, M., Kadri, H.: Learning in RKHM: a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic twist for kernel machines. In: Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS) (2023) Hasimoto et al. [2023b] Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Ishikawa, I., Ikeda, M., Komura, F., Katsura, T., Kawahara, Y.: Reproducing kernel Hilbert C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-module and kernel mean embeddings. Journal of Machine Learning Research 22(267), 1–56 (2021) Hasimoto et al. [2023a] Hasimoto, Y., Ikeda, M., Kadri, H.: Learning in RKHM: a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic twist for kernel machines. In: Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS) (2023) Hasimoto et al. [2023b] Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hasimoto, Y., Ikeda, M., Kadri, H.: Learning in RKHM: a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic twist for kernel machines. In: Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS) (2023) Hasimoto et al. [2023b] Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Hashimoto, Y., Ishikawa, I., Ikeda, M., Komura, F., Katsura, T., Kawahara, Y.: Reproducing kernel Hilbert C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-module and kernel mean embeddings. Journal of Machine Learning Research 22(267), 1–56 (2021) Hasimoto et al. [2023a] Hasimoto, Y., Ikeda, M., Kadri, H.: Learning in RKHM: a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic twist for kernel machines. In: Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS) (2023) Hasimoto et al. [2023b] Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hasimoto, Y., Ikeda, M., Kadri, H.: Learning in RKHM: a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic twist for kernel machines. In: Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS) (2023) Hasimoto et al. [2023b] Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Hasimoto, Y., Ikeda, M., Kadri, H.: Learning in RKHM: a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic twist for kernel machines. In: Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS) (2023) Hasimoto et al. [2023b] Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Hasimoto, Y., Ikeda, M., Kadri, H.: Deep learning with kernels through RKHM and the Perron-Frobenius operator. In: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS) (2023) [7] Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Hashimoto, Y., Ikeda, M., Kadri, H.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebraic machine learning: Moving in a new direction. arXiv preprint arXiv: 2402.02637 (2024) Hashimoto et al. [2022] Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Hashimoto, Y., Wang, Z., Matsui, T.: C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra net: a new approach generalizing neural network parameters to C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. In: Proceedings of the 39th International Conference on Machine Learning (ICML) (2022) Dong et al. [2020] Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Dong, X., Yu, Z., Cao, W., Shi, Y., Ma, Q.: A survey on ensemble learning. Frontiers of Computer Science 14, 241–258 (2020) Ganaie et al. [2022] Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Ganaie, M.A., Hu, M., Malik, A.K., Tanveer, M., Suganthan, P.N.: Ensemble deep learning: A review. Engineering Applications of Artificial Intelligence 115, 105151 (2022) Fei-Fei et al. [2006] Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(4), 594–611 (2006) Lake et al. [2011] Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Lake, B., Salakhutdinov, R., Gross, J., Tenenbaum, J.: One shot learning of simple visual concepts. In: Proceedings of the Annual Meeting of the Cognitive Science Society, vol. 33 (2011) Hataya and Hashimoto [2023] Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Hataya, R., Hashimoto, Y.: Noncommutative C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra Net: Learning Neural Networks with Powerful Product Structure in C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. arXiv preprint arXiv: 2302.01191 (2023) Le Cun et al. [1998] Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Le Cun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) Shirakawa et al. [2021] Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Shirakawa, T., Ueda, H., Yunoki, S.: Automatic quantum circuit encoding of a given arbitrary quantum state. arXiv preprint arXiv:2112.14524 (2021) Vanhaesebrouck et al. [2017] Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Vanhaesebrouck, P., Bellet, A., Tommasi, M.: Decentralized collaborative learning of personalized models over networks. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) (2017) Bellet et al. [2018] Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Bellet, A., Guerraoui, R., Taziki, M., Tommasi, M.: Personalized and private peer-to-peer machine learning. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) (2018) Elman [1990] Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Elman, J.L.: Finding structure in time. Cognitive Science 14(2), 179–211 (1990) Xiao et al. [2017] Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Xiao, H., Rasul, K., Vollgraf, R.: Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv (2017) [20] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., Ha, D.: Deep Learning for Classical Japanese Literature. arXiv preprint arXiv: 1812.01718 (2018) Boumal [2023] Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023) Kingma and Ba [2015] Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Kingma, D.P., Ba, J.L.: Adam: a Method for Stochastic Optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015) Havlíček et al. [2019] Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Havlíček, V., Córcoles, A.D., Temme, K., Harrow, A.W., Kandala, A., Chow, J.M., Gambetta, J.M.: Supervised learning with quantum-enhanced feature spaces. Nature 567(7747), 209–212 (2019) Kusumoto et al. [2021] Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M., Negoro, M.: Experimental quantum kernel trick with nuclear spins in a solid. npj Quantum Information 7(1), 94 (2021) Placidi et al. [2023] Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Placidi, L., Hataya, R., Mori, T., Aoyama, K., Morisaki, H., Mitarai, K., Fujii, K.: Mnisq: A large-scale quantum circuit dataset for machine learning on/for quantum computers in the NISQ era. arXiv preprint arXiv:2306.16627 (2023) Holmes et al. [2023] Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Holmes, Z., Coble, N.J., Sornborger, A.T., Subaşi, Y.: Nonlinear transformations in quantum computation. Physical Review Research 5, 013105 (2023) Nishimori and Akaho [2005] Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the stiefel manifold. Neurocomputing 67, 106–135 (2005) Wisdom et al. [2016] Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Wisdom, S., Powers, T., Hershey, J., Le Roux, J., Atlas, L.: Full-capacity unitary recurrent neural networks. Proceedings in the 30th Conference on neural information processing systems (NeurIPS) (2016) Li et al. [2020] Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Li, J., Li, F., Todorovic, S.: Efficient riemannian optimization on the stiefel manifold via the cayley transform. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020) Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Proceedings in the 33th Conference on Neural Information Processing Systems (NeurIPS) (2019) Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)
- Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. Journal of Machine Learning Research 12, 2825–2830 (2011)