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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.
