1 research outputs found
Kernel interpolation with continuous volume sampling
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