Accurate error bounds for the eigenvalues of the kernel matrix

The eigenvalues of the kernel matrix play an important role in a number of kernel methods, in particular, in kernel principal component analysis. It is well known that the eigenvalues of the kernel matrix converge as the number of samples tends to infinity. We derive probabilistic finite sample size bounds on the approximation error of individual eigenvalues which have the important property that the bounds scale with the eigenvalue under consideration, reflecting the actual behavior of the approximation errors as predicted by asymptotic results and observed in numerical simulations. Such scaling bounds have so far only been known for tail sums of eigenvalues. Asymptotically, the bounds presented here have a slower than stochastic rate, but the number of sample points necessary to make this disadvantage noticeable is often unrealistically large. Therefore, under practical conditions, and for all but the largest few eigenvalues, the bounds presented here form a significant improvement over existing non-scaling bounds.