Positively Weighted Kernel Quadrature via Subsampling

Abstract

We study kernel quadrature rules with positive weights for probability measures on general domains. Our theoretical analysis combines the spectral properties of the kernel with random sampling of points. This results in effective algorithms to construct kernel quadrature rules with positive weights and small worst-case error. Besides additional robustness, our numerical experiments indicate that this can achieve fast convergence rates that compete with the optimal bounds in well-known examples.

Markov Chain Approximations to Stochastic Differential Equations by Recombination on Lattice Trees

“…Remark 4. There are other recombination algorithms [20,21,22,37,43,53] besides Algorithm 2 from [6]. However, what makes Algorithm 2 especially attractive in the current situation is that it can take as input a sequence of points -in our case, points sampled uniformly on the lattice -and the moments that need to be matched.…”

“…and then reduce the discrete probability measure obtained by (iii) further by using any of the recombination algorithms in [20,21,22,37,43,53].…”

“…However, the proof of Carathéodory's Theorem is not constructive and the design of algorithms that carry out such a measure construction efficiently is still the subject of recent research; e.g. just over the last ten years the articles [6,20,21,22,37,43,53] provide novel algorithms. Recombination has already been used in the context of SDE simulations: in order to make "Cubature on Wiener Space" [42] efficient, a recombination step is applied iteratively over time, similar in spirit to our construction; see [37] for a discussion on how powerful this can be for cubature methods on Wiener space.…”

Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures

“…Empirically, this approach turns out to work well already for "reasonable" magnitudes of N [24,27]. The aim of this article is to fill this gap and provide theoretical guarantees for the number of samples N for which this approach leads with high probability to a successful cubature construction; that is to provide a quantitative version of Proposition 1 that applies to common cases.…”

“…Arguably the most famous applications concerns the case when X is a subset of R d and F is the linear space of polynomials up to a certain degree, that is F is spanned by monomials up to a certain degree. However, more recent applications include the case when X is a space of paths and F is spanned by iterated Ito-Stratonovich integrals [39], or kernel quadrature [33,27] where X is a set that carries a positie definite kernel and F is a subset of the associated reproducing kernel Hilber space that is spanned by eigenfunctions of the integral operator induced by a kernel.…”