Universal kernels via harmonic analysis on Riemannian symmetric spaces

TL;DR

This paper develops tools based on harmonic analysis to study the universality of kernels on Riemannian symmetric spaces, extending kernel theory to non-Euclidean domains and validating recent kernels for manifold data.

Contribution

It introduces fundamental tools for analyzing kernel universality on Riemannian symmetric spaces and proves the universality of several recent kernels in this setting.

Findings

Established harmonic analysis tools for kernel universality on symmetric spaces.

Proved universality of recent positive definite kernels on Riemannian symmetric spaces.

Provided theoretical justification for kernel use in manifold-valued data applications.

Abstract

The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in machine learning. In this work, we establish fundamental tools for investigating universality properties of kernels in Riemannian symmetric spaces, thereby extending the study of this important topic to kernels in non-Euclidean domains. Moreover, we use the developed tools to prove the universality of several recent examples from the literature on positive definite kernels defined on Riemannian symmetric spaces, thus providing theoretical justification for their use in applications involving manifold-valued data.