Kernel interpolation with continuous volume sampling

Ayoub Belhadji, Rémi Bardenet, Pierre Chainais

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 quadrature or kernel interpolation
from discrete samples. Furthermore, RKHSs offer a convenient mathematical and computational
framework, connecting the discrete and continuous worlds. 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 only 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.