Sobolev Spaces, Kernels and Discrepancies over Hyperspheres

Abstract

This work extends analytical foundations for kernel methods beyond the usual Euclidean manifold. Specifically, we characterise the smoothness of the native spaces (reproducing kernel Hilbert spaces) that are reproduced by geodesically isotropic kernels in the hyperspherical context. Our results are relevant to several areas of machine learning; we focus on their consequences for kernel cubature, determining the rate of convergence of the worst case error, and expanding the applicability of cubature algorithms based on Stein's method. First, we introduce a characterisation of Sobolev spaces on the $d$-dimensional sphere based on the Fourier--Schoenberg sequences associated with a given kernel. Such sequences are hard (if not impossible) to compute analytically on $d$-dimensional spheres, but often feasible over Hilbert spheres, where $d = \infty$. Second, we circumvent this problem by finding a projection operator that allows us to map from Hilbert spheres to finite-dimensional spheres. Our findings are illustrated for selected parametric families of kernel.