Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures

Abstract

Given a probability measure µ on a set X and a vector-valued function ϕ, a common problem is to construct a discrete probability measure on X such that the push-forward of these two probability measures under ϕ is the same. This construction is at the heart of numerical integration methods that run under various names such as quadrature, cubature, or recombination. A natural approach is to sample points from µ until their convex hull of their image under ϕ includes the mean of ϕ. Here we analyze the computational complexity of this approach when ϕ exhibits a graded structure by using so-called hypercontractivity. The resulting theorem not only covers the classical cubature case of multivariate polynomials, but also integration on pathspace, as well as kernel quadrature for product measures.

Sampling-based Nyström Approximation and Kernel Quadrature

“…Although there is a randomized algorithm for constructing the Q n stated in the above theorem [22, Algorithm 2 with modification], it has two issues; it requires exact values of µ(ϕ i ) (and µ( k − k app )) and its computational complexity has no useful upper bound except some cases with special structure such as a product kernel with a product measure [23]. This said, we can deduce updated convergence results for outputs of the algorithm as in Remark 1.…”