A close look at the entropy numbers of the unit ball of the Reproducing Hilbert Space of isotropic positive definite kernels

TL;DR

This paper provides precise upper and lower bounds for the covering numbers of the unit ball in RKHSs generated by isotropic positive definite kernels on spheres, with explicit constants and asymptotic behavior.

Contribution

It introduces accurate bounds with explicit constants for the covering numbers of RKHS unit balls on spheres, detailing their dependence on dimension and kernel coefficients.

Findings

Explicit bounds for covering numbers with constants

Asymptotic behavior depending on dimension and kernel coefficients

Improved understanding of RKHS complexity on spheres

Abstract

We present accurate upper and lower bounds for the covering numbers, with explicit constants, of the unit ball for two general classes of Reproducing Kernel Hilbert Space (RKHS) on the unit sphere of $\mathbb{R}^{d+1}$. In both classes, the RKHS is generated by an isotropic continuous positive definite kernel. The upper and lower bounds we present carry precise information about the asymptotic constants, depending on the dimension of the sphere and the monotonic behavior of the Schoenberg/Fourier coefficients of the isotropic kernel.