Statistical Properties of Kernel Principal Component Analysis

Zwald, L; Bousquet, O; Blanchard , G

doi:10.1007/978-3-540-27819-1_41

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Statistical Properties of Kernel Principal Component Analysis

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Bousquet, O
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Zwald, L., Bousquet, O., & Blanchard, G. (2004). Statistical Properties of Kernel Principal Component Analysis. In J. Shawe-Taylor, & Y. Singer (Eds.), Learning Theory: 17th Annual Conference on Learning Theory, COLT 2004, Banff, Canada, July 1-4 (pp. 594-608). Berlin, Germany: Springer.

Cite as: https://hdl.handle.net/21.11116/0000-0005-5340-5

Abstract

We study the properties of the eigenvalues of Gram matrices in a non-asymptotic setting. Using local Rademacher averages, we provide data-dependent and tight bounds for their convergence towards eigenvalues of the corresponding kernel operator. We perform these computations in a functional analytic framework which allows to deal implicitly with reproducing kernel Hilbert spaces of infinite dimension. This can have applications to various kernel algorithms, such as Support Vector Machines (SVM). We focus on Kernel Principal Component Analysis (KPCA) and, using such techniques, we obtain sharp excess risk bounds for the reconstruction error. In these bounds, the dependence on the decay of the spectrum and on the closeness of successive eigenvalues is made explicit.