Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures

TL;DR

This paper analyzes the computational complexity of constructing discrete measures for numerical integration using hypercontractivity, covering classical cubature, pathspace integration, and kernel quadrature.

Contribution

It introduces a hypercontractivity-based analysis framework for randomized multivariate cubatures, extending classical and modern integration methods.

Findings

Provides complexity bounds for hypercontractive functions in cubature

Extends analysis to pathspace and kernel quadrature cases

Unifies various numerical integration approaches under a common framework

Abstract

Given a probability measure $\mu$ on a set $\mathcal{X}$ and a vector-valued function $\varphi$, a common problem is to construct a discrete probability measure on $\mathcal{X}$ such that the push-forward of these two probability measures under $\varphi$ 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 $\mu$ until their convex hull of their image under $\varphi$ includes the mean of $\varphi$. Here we analyze the computational complexity of this approach when $\varphi$ 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.