Non-asymptotic spectral bounds on the -entropy of kernel classes

Abstract

Let K: be a continuous Mercer kernel defined on a compact subset of R^n and H_K be the reproducing kernel Hilbert space (RKHS) associated with K. Given a finite measure on , we investigate upper and lower bounds on the -entropy of the unit ball of H_K in the space L_p(). This topic is an important direction in the modern statistical theory of kernel-based methods. We prove sharp upper and lower bounds for p [1,+]. For p [1,2], the upper bounds are determined solely by the eigenvalue behaviour of the corresponding integral operator _ K(, y)( y)d( y). In constrast, for p>2, the bounds additionally depend on the convergence rate of the truncated Mercer series to the kernel K in the L_p()-norm. We discuss a number of consequences of our bounds and show that they are substantially tighter than previous bounds for general kernels. Furthermore, for specific cases, such as zonal kernels and the Gaussian kernel on a box, our bounds are asymptotically tight as +0.

Tasks

Reproductions