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

Let K: boldsymbolOmega× boldsymbolOmega be a continuous Mercer kernel defined on a compact subset of mathbb Rn and HK be the reproducing kernel Hilbert space (RKHS) associated with K. Given a finite measure nu on boldsymbolOmega, we investigate upper and lower bounds on the varepsilon-entropy of the unit ball of HK in the space Lp(nu). 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 φ→ int_boldsymbolOmega K(·,mathbf y)φ(mathbf y)dnu(mathbf 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 Lp(nu)-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 varepsilon→ +0.