Positively Weighted Kernel Quadrature via Subsampling

Satoshi Hayakawa, Harald Oberhauser, Terry Lyons

Keywords: kernel quadrature, recombination, reproducing kernel Hilbert space, Nyström approximation

Abstract: We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d. samples from the underlying measure and evaluation of the kernel and that result in a small worst-case error. In addition to our theoretical results and the benefits resulting from convex weights, our experiments indicate that this construction can compete with the optimal bounds in well-known examples.

TL;DR: We give an efficient algorithm and theoretical guarantee for a kernel quadrature rule with convex weights.

Community Implementations: [ 2 code implementations](https://www.catalyzex.com/paper/arxiv:2107.09597/code)