On the Eigenspectrum of the Gram matrix and the generalisation error of kernel PCA

Abstract

In this paper we analyze the relationships between the eigenvalues of the m x m Gram matrix K for a kernel k(.,.) corresponding to a sample x1,...,xm drawn from a density p(x) and the eigenvalues of the corresponding continuous eigenproblem. We bound the differences between the two spectra and provide a performance bound on kernel PCA showing that we can expect good performance even in very high dimensional feature spaces provided the sample eigenvalues fall sufficiently quickly.

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