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,...,x_m drawn from a density p(x) and the eigenvalues of the corresponding continuous eigenproblem is analyzed. The differences between the two spectra are bounded and a performance bound on kernel principal component analysis (PCA) is provided showing that good performance can be expected even in very-high-dimensional feature spaces provided the sample eigenvalues fall sufficiently quickly.