Monte Carlo Cubature Construction

TL;DR

This paper introduces a novel Monte Carlo-based method for constructing cubature formulas for probability measures, enabling numerical integration and data compression without extensive prior knowledge of the measure.

Contribution

It presents a new approach to build cubature formulas using i.i.d. samples and test function means, applicable beyond traditional domains like hypercubes and spheres.

Findings

Constructs cubature formulas from samples and test function means.

Applicable to general probability measures, not limited to specific domains.

Provides a data compression technique using the same framework.

Abstract

In numerical integration, cubature methods are effective, especially when the integrands can be well-approximated by known test functions, such as polynomials. However, the construction of cubature formulas has not generally been known, and existing examples only represent the particular domains of integrands, such as hypercubes and spheres. In this study, we show that cubature formulas can be constructed for probability measures provided that we have an i.i.d. sampler from the measure and the mean values of given test functions. Moreover, the proposed method also works as a means of data compression, even if sufficient prior information of the measure is not available.