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Abstrakty
Motivated by applications, we consider new operator-theoretic approaches to conditional mean embedding (CME). Our present results combine a spectral analysis-based optimization scheme with the use of kernels, stochastic processes, and constructive learning algorithms. For initially given non-linear data, we consider optimization-based feature selections. This entails the use of convex sets of kernels in a construction of optimal feature selection via regression algorithms from learning models. Thus, with initial inputs of training data (for a suitable learning algorithm), each choice of a kernel K in turn yields a variety of Hilbert spaces and realizations of features. A novel aspect of our work is the inclusion of a secondary optimization process over a specified convex set of positive definite kernels, resulting in the determination of “optimal” feature representations.
Czasopismo
Rocznik
Tom
Strony
79--103
Opis fizyczny
Bibliogr. 30 poz., tab., wykr.
Twórcy
autor
- Department of Mathematics, The University of Iowa, Iowa City, IA 52242–1419, U.S.A.
autor
- Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026, U.S.A.
autor
- Mathematical Reviews, 416–4th Street Ann Arbor, MI 48103–4816, U.S.A.
Bibliografia
- [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), no. 3, 337–404.
- [2] J. Cerviño, J.A. Bazerque, M. Calvo-Fullana, A. Ribeiro, Multi-task reinforcement learning in reproducing kernel Hilbert spaces via cross-learning, IEEE Trans. Signal Process. 69 (2021), 5947–5962.
- [3] N. Dunford, J.T. Schwartz, Linear Operators. Part II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988.
- [4] S. Grünewälder, G. Lever, L. Baldassarre, S. Patterson, A. Gretton, M. Pontil, Conditional mean embeddings as regressors, [in:] Proceedings of the 29th International Conference on International Conference on Machine Learning, Omnipress, Madison, WI, USA, 2012, ICML’12, 1803–1810.
- [5] D. He, J. Cheng, K. Xu, High-dimensional variable screening through kernel-based conditional mean dependence, J. Statist. Plann. Inference 224 (2023), 27–41.
- [6] P. Jorgensen, F. Tian, Non-commutative Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
- [7] P. Jorgensen, F. Tian, Decomposition of Gaussian processes, and factorization of positive definite kernels, Opuscula Math. 39 (2019), no. 4, 497–541.
- [8] P. Jorgensen, F. Tian, Realizations and factorizations of positive definite kernels, J. Theoret. Probab. 32 (2019), no. 4, 1925–1942.
- [9] P. Jorgensen, F. Tian, Sampling with positive definite kernels and an associated dichotomy, Adv. Theor. Math. Phys. 24 (2020), no. 1, 125–154.
- [10] P. Jorgensen, F. Tian, Reproducing kernels: harmonic analysis and some of their applications, Appl. Comput. Harmon. Anal. 52 (2021), 279–302.
- [11] P. Jorgensen, F. Tian, Infinite-dimensional Analysis – Operators in Hilbert Space; Stochastic Calculus via Representations, and Duality Theory, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2021.
- [12] P. Jorgensen, F. Tian, Reproducing kernels and choices of associated feature spaces, in the form of L2-spaces, J. Math. Anal. Appl. 505 (2022), no. 2, 125535.
- [13] P.E.T. Jorgensen, M.-S. Song, J. Tian, Positive definite kernels, algorithms, frames, and approximations, (2021), arXiv:2104.11807.
- [14] I. Klebanov, I. Schuster, T.J. Sullivan, A rigorous theory of conditional mean embeddings, SIAM J. Math. Data Sci. 2 (2020), no. 3, 583–606.
- [15] T. Lai, Z. Zhang, Y. Wang, A kernel-based measure for conditional mean dependence, Comput. Statist. Data Anal. 160 (2021), Paper no. 107246.
- [16] T. Lai, Z. Zhang, Y. Wang, L. Kong, Testing independence of functional variables by angle covariance, J. Multivariate Anal. 182 (2021), Paper no. 104711.
- [17] Y.J. Lee, C.A. Micchelli, J. Yoon, On multivariate discrete least squares, J. Approx. Theory 211 (2016), 78–84.
- [18] G. Lever, J. Shawe-Taylor, R. Stafford, C. Szepesvári, Compressed conditional mean embeddings for model-based reinforcement learning, AAAI’16: Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence (2016), 1779–1787.
- [19] D.K. Lim, N.U. Rashid, J.G. Ibrahim, Model-based feature selection and clustering of RNA-seq data for unsupervised subtype discovery, Ann. Appl. Stat. 15 (2021), no. 1, 481–508.
- [20] C.-K. Lu, P. Shafto, Conditional deep Gaussian processes: Multi-fidelity kernel learning, Entropy 2021, 23(11), 1545.
- [21] E. Mehmanchi, A. Gómez, O.A. Prokopyev, Solving a class of feature selection problems via fractional 0–1 programming, Ann. Oper. Res. 303 (2021), 265–295.
- [22] C.A. Micchelli, M. Pontil, Q. Wu, D.-X. Zhou, Error bounds for learning the kernel, Anal. Appl. (Singap.) 14 (2016), no. 6, 849–868.
- [23] P. Niyogi, S. Smale, S. Weinberger, A topological view of unsupervised learning from noisy data, SIAM J. Comput. 40 (2011), no. 3, 646–663.
- [24] J. Park, K. Muandet, A measure-theoretic approach to kernel conditional mean embeddings, arXiv:2002.03689.
- [25] S. Ray Chowdhury, R. Oliveira, F. Ramos, Active learning of conditional mean embeddings via Bayesian optimisation, [in:] J. Peters, D. Sontag (eds) Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI), PMLR, 2020, volume 124 of Proceedings of Machine Learning Research, 1119–1128.
- [26] S. Smale, Y. Yao, Online learning algorithms, Found. Comput. Math. 6 (2006), 145–170.
- [27] S. Smale, D.-X. Zhou, Geometry on probability spaces, Constr. Approx. 30 (2009), 311–323.
- [28] P. Xu, Y. Wang, X. Chen, Z. Tian, COKE: communication-censored decentralized kernel learning, J. Mach. Learn. Res. 22 (2021), Paper no. 196.
- [29] Y. Zhang, Y.-C. Chen, Kernel smoothing, mean shift, and their learning theory with directional data, J. Mach. Learn. Res. 22 (2021), Paper no. 154.
- [30] P. Zhao, L. Lai, Minimax rate optimal adaptive nearest neighbor classification and regression, IEEE Trans. Inform. Theory 67 (2021), no. 5, 3155–3182.
Uwagi
Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-214208c7-ae7e-46c6-ac50-d34ca9432a8c