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 analyse 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.