Relative concentration bounds for the spectrum of kernel matrices

In this paper we study some concentration properties of the kernel matrix associated with a kernel function. More specifically, we derive concentration inequalities for the spectrum of a kernel matrix, quantifying its deviation with respect to the spectrum of an associated integral operator. The main difference with most results in the literature is that we do not assume the positive definiteness of the kernel. Instead, we introduce Sobolev-type hypotheses on the regularity of the kernel. We show how these assumptions are well-suited to the study of kernels depending only on the distance between two points in a metric space, in which case the regularity only depends on the decay of the eigenvalues. This is connected with random geometric graphs, which we study further, giving explicit formulas for the spectrum and its fluctuations.