Sobolev Spaces, Kernels and Discrepancies over Hyperspheres

TL;DR

This paper develops a theoretical framework for kernel methods on hyperspheres, characterizing associated Sobolev spaces and analyzing convergence rates for kernel cubature, with implications for Stein's method-based algorithms.

Contribution

It provides a novel characterization of Sobolev spaces on hyperspheres using Fourier--Schoenberg sequences and introduces a projection operator to facilitate analysis.

Findings

Characterization of Sobolev spaces on hyperspheres via Fourier--Schoenberg sequences.

A projection operator enabling Fourier mapping from Hilbert to finite-dimensional hyperspheres.

Analysis of convergence rates for kernel cubature algorithms.

Abstract

This work provides theoretical foundations for kernel methods in the hyperspherical context. Specifically, we characterise the native spaces (reproducing kernel Hilbert spaces) and the Sobolev spaces associated with kernels defined over hyperspheres. Our results have direct 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. We first introduce a suitable characterisation on Sobolev spaces on the $d$-dimensional hypersphere embedded in $(d+1)$-dimensional Euclidean spaces. Our characterisation is 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. We circumvent this problem by finding a projection operator that allows to Fourier mapping from Hilbert into finite dimensional hyperspheres. We illustrate our findings through some parametric families of kernels.