Entropy numbers of Reproducing Hilbert Space of zonal positive definite kernels on compact two-point homogeneous spaces

TL;DR

This paper provides detailed estimates for the covering numbers of RKHS unit balls on compact two-point homogeneous spaces, extending previous sphere results and applying to specific kernels like the spherical Gaussian.

Contribution

It introduces new bounds for covering numbers of RKHSs on general compact homogeneous spaces using Schoenberg series, extending known sphere results.

Findings

Derived asymptotic constants depending on manifold dimension

Extended covering number estimates to general homogeneous spaces

Applied bounds to spherical Gaussian kernel

Abstract

We present estimates for the covering numbers of the unit ball of Reproducing Kernel Hilbert Spaces (RKHSs) of functions on $M^d$ a d-dimensional compact two-point homogeneous space. The RKHS is generated by a continuous zonal/isotropic positive definite kernel. We employ the representation in terms of the Schoenberg/Fourier series expansion for continuous isotropic positive definite kernels, given in terms of a family of orthogonal polynomials on $M^d$. The bounds we present carry accurate information about the asymptotic constants depending on the dimension of the manifold and the decay or growth rate of the coefficients of the kernel. The results we present extend the estimates previously known for continuous isotropic positive definite kernels on the d-dimensional unit sphere. We present the weak asymptotic equivalence for the order of the growth of covering numbers associated to kernels on $M^d$ with a convergent geometric sequence of coefficients. We apply our estimates in order to present a bound for the covering numbers of the spherical Gaussian kernel, and to present bounds for formal examples on $M^d$.