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

Michael Kazhdan (1) , Gurprit Singh (2) , Adrien Pilleboue (2) , David Coeurjolly (3) , Victor Ostromoukhov (2)

1 JHU - Johns Hopkins University
2 R3AM - Rendu Réaliste pour la Réalité Augmentée Mobile
3 M2DisCo - Geometry Processing and Constrained Optimization

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. 2 Problem Statmement We begin by reviewing some basic concepts from Monte Carlo integration. Next, we present a formal definition of homogeneity. And finally, we formulate the generalized problem statement.

Domaines: Traitement des images [eess.IV]; Géométrie algorithmique [cs.CG]

Michael Kazhdan, Gurprit Singh, Adrien Pilleboue, David Coeurjolly, Victor Ostromoukhov. Variance Analysis for Monte Carlo Integration: A Representation-Theoretic Perspective. [Research Report] LIRIS UMR CNRS 5205. 2015. ⟨hal-01259838⟩