Kernel interpolation with continuous volume sampling
arXiv:2002.09677
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.
References in corpus (9)
- Determinantal Processes and Independence
- Leveraged volume sampling for linear regression
- Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees
- Optimally-Weighted Herding is Bayesian Quadrature
- Sequential Kernel Herding: Frank-Wolfe Optimization for Particle Filtering
- Complexity of Oscillatory Integration for Univariate Sobolev Spaces
- A determinantal point process for column subset selection
- On two ways to use determinantal point processes for Monte Carlo integration
- Control Functionals for Quasi-Monte Carlo Integration