On the eigenspectrum of the Gram matrix and the generalization error of kernel-PCA

In this paper the relationships between the eigenvalues of the m × m Gram matrix K for a kernel κ(·, ·) corresponding to a sample x1,..., xm drawn from a density p(x) and the eigenvalues of the corresponding continuous eigenproblem is analysed. The differences between the two spectra are bounded and a performance bound on kernel PCA is provided showing that good performance can be expected even in very high dimensional feature spaces provided the sample eigenvalues fall sufficiently quickly