Kernel interpolation with continuous volume sampling

TL;DR

This paper introduces continuous volume sampling (VS), a novel method for selecting nodes in kernel methods, providing near-optimal theoretical bounds for interpolation and quadrature applicable to any Mercer kernel, with practical advantages for sampling.

Contribution

The paper presents the theoretical development of continuous volume sampling (VS), extending discrete VS to continuous node selection in kernel methods, with bounds valid for all Mercer kernels.

Findings

Provides near-optimal bounds for interpolation and quadrature using VS.

VS applies to any Mercer kernel and depends on the spectrum of the associated operator.

Evaluation of VS only requires pointwise kernel evaluations, facilitating MCMC sampling.

Abstract

A fundamental task in kernel methods is to pick nodes and weights, so as to approximate a given function from an RKHS by the weighted sum of kernel translates located at the nodes. This is the crux of kernel density estimation, kernel quadrature, or interpolation from discrete samples. Furthermore, RKHSs offer a convenient mathematical and computational framework. We introduce and analyse continuous volume sampling (VS), the continuous counterpart -- for choosing node locations -- of a discrete distribution introduced in (Deshpande & Vempala, 2006). Our contribution is theoretical: we prove almost optimal bounds for interpolation and quadrature under VS. While similar bounds already exist for some specific RKHSs using ad-hoc node constructions, VS offers bounds that apply to any Mercer kernel and depend on the spectrum of the associated integration operator. We emphasize that, unlike previous randomized approaches that rely on regularized leverage scores or determinantal point processes, evaluating the pdf of VS only requires pointwise evaluations of the kernel. VS is thus naturally amenable to MCMC samplers.