Symmetry Exploits for Bayesian Cubature Methods

TL;DR

This paper enhances Bayesian cubature methods by exploiting symmetry properties to reduce computational costs, extending previous work to non-symmetric measures and multiple Bayesian approaches.

Contribution

It introduces new symmetry exploitation techniques for Bayesian cubature, including non-symmetric measures and multi-output methods, improving efficiency.

Findings

Symmetry exploitation reduces computational cost significantly.

Extensions to non-symmetric measures are feasible.

Multiple Bayesian cubature methods benefit from symmetry techniques.

Abstract

Bayesian cubature provides a flexible framework for numerical integration, in which a priori knowledge on the integrand can be encoded and exploited. This additional flexibility, compared to many classical cubature methods, comes at a computational cost which is cubic in the number of evaluations of the integrand. It has been recently observed that fully symmetric point sets can be exploited in order to reduce - in some cases substantially - the computational cost of the standard Bayesian cubature method. This work identifies several additional symmetry exploits within the Bayesian cubature framework. In particular, we go beyond earlier work in considering non-symmetric measures and, in addition to the standard Bayesian cubature method, present exploits for the Bayes-Sard cubature method and the multi-output Bayesian cubature method.