Variance Analysis for Monte Carlo Integration: A Representation-Theoretic Perspective

TL;DR

This paper offers a representation-theoretic framework for analyzing the variance of Monte Carlo integration across different geometric spaces, unifying previous results into a general theory.

Contribution

It introduces a novel, unified approach using representation theory to derive closed-form variance expressions for Monte Carlo integration on various manifolds.

Findings

Unified variance formulas for torus, sphere, and Euclidean space

Representation theory provides a general framework for Monte Carlo variance analysis

Closed-form solutions facilitate better understanding of Monte Carlo error behavior

Abstract

In this report, we revisit the work of Pilleboue et al. [2015], providing a representation-theoretic derivation of the closed-form expression for the expected value and variance in homogeneous Monte Carlo integration. We show that the results obtained for the variance estimation of Monte Carlo integration on the torus, the sphere, and Euclidean space can be formulated as specific instances of a more general theory. We review the related representation theory and show how it can be used to derive a closed-form solution.