[21] Peter L. Bartlett, Philip M. Long, Gábor Lugosi, and Alexander Tsigler. Benign overfitting in linear regression. Proceedings of the National Academy of Sciences, 117(48):30063-30070, 2020.
[22] Neil Rohit Mallinar, James B Simon, Amirhesam Abedsoltan, Parthe Pandit, Misha Belkin, and Preetum Nakkiran. Benign, tempered, or catastrophic: Toward a refined taxonomy of overfitting. In Proceedings of Advances in Neural Information Processing Systems (NeurIPS) 37, 2022.
[23] Jaeseong Heo. Reproducing kernel Hilbert  $C^*$ -modules and kernels associated with cocycles. Journal of Mathematical Physics, 49(10):103507, 2008.
[24] Shigeru Itoh. Reproducing kernels in modules over  $C^*$ -algebras and their applications. Journal of Mathematics in Nature Science, pages 1-20, 1990.
[25] Mohammad S. Moslehian. Vector-valued reproducing kernel Hilbert  $C^*$ -modules. Complex Analysis and Operator Theory, 16(1):Paper No. 2, 2022.
[26] Yuka Hashimoto, Isao Ishikawa, Masahiro Ikeda, Fuyuta Komura, Takeshi Katsura, and Yoshi-nobu Kawahara. Reproducing kernel Hilbert  $C^*$ -module and kernel mean embeddings. Journal of Machine Learning Research, 22(267):1-56, 2021.
[27] Yuka Hashimoto, Masahiro Ikeda, and Hachem Kadri. Learning in RKHM: a  $C^*$ -algebraic twist for kernel machines. In Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS), 2023.
[28] Yoshinobu Kawahara. Dynamic mode decomposition with reproducing kernels for Koopman spectral analysis. In Advances in Neural Information Processing Systems (NIPS) 29, pages 911-919, 2016.
[29] Isao Ishikawa, Keisuke Fujii, Masahiro Ikeda, Yuka Hashimoto, and Yoshinobu Kawahara. Metric on nonlinear dynamical systems with Perron-Frobenius operators. In Advances in Neural Information Processing Systems (NeurIPS) 31, pages 2856-2866, 2018.
[30] Dimitrios Giannakis and Suddhasattwa Das. Extraction and prediction of coherent patterns in incompressible flows through space-time Koopman analysis. Physica D: Nonlinear Phenomena, 402:132211, 2020.
[31] Yuka Hashimoto, Isao Ishikawa, Masahiro Ikeda, Yoichi Matsuo, and Yoshinobu Kawahara. Krylov subspace method for nonlinear dynamical systems with random noise. Journal of Machine Learning Research, 21(172):1-29, 2020.
[32] Ryuichiro Hataya and Yuka Hashimoto. Noncommutative  $C^*$ -algebra net: Learning neural networks with powerful product structure in  $C^*$ -algebra. arXiv: 2302.01191, 2023.
[33] E. Christopher Lance. Hilbert  $C^*$ -modules - a Toolkit for Operator Algebraists. London Mathematical Society Lecture Note Series, vol. 210. Cambridge University Press, 1995.
[34] Gerard J. Murphy.  $C^*$ -Algebras and Operator Theory. Academic Press, 1990.
[35] Yuka Hashimoto, Isao Ishikawa, Masahiro Ikeda, Fuyuta Komura, Takeshi Katsura, and Yoshi-nobu Kawahara. Analysis via orthonormal systems in reproducing kernel hilbert  $c^*$ -modules and applications. arXiv: 2003.00738, 2020.
[36] Mauricio A Álvarez, Lorenzo Rosasco, Neil D Lawrence, et al. Kernels for vector-valued functions: A review. Foundations and Trends® in Machine Learning, 4(3):195-266, 2012.
[37] Ali Rahimi and Benjamin Recht. Random features for large-scale kernel machines. In Proceedings of the Advances in Neural Information Processing Systems 20 (NIPS), 2007.
[38] Yuka Hashimoto, Zhao Wang, and Tomoko Matsui.  $C^*$ -algebra net: a new approach generalizing neural network parameters to  $C^*$ -algebra. In Proceedings of the 39th International Conference on Machine Learning (ICML), 2022.
[39] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278-2324, 1998.