Optimal projection to improve parametric importance sampling in high dimension

TL;DR

This paper introduces a dimension-reduction method for importance sampling in high dimensions by identifying optimal directions through eigenvector analysis, significantly enhancing estimation accuracy.

Contribution

It provides a theoretical framework for selecting optimal projection directions based on eigenvalues, improving importance sampling performance in high-dimensional settings.

Findings

Optimal directions are eigenvectors of the covariance matrix associated with extreme eigenvalues.

Estimations remain accurate even with simple empirical covariance estimators.

Numerical experiments confirm the method's effectiveness as dimension increases.

Abstract

In this paper we propose a dimension-reduction strategy in order to improve the performance of importance sampling in high dimension. The idea is to estimate variance terms in a small number of suitably chosen directions. We first prove that the optimal directions, i.e., the ones that minimize the Kullback--Leibler divergence with the optimal auxiliary density, are the eigenvectors associated to extreme (small or large) eigenvalues of the optimal covariance matrix. We then perform extensive numerical experiments that show that as dimension increases, these directions give estimations which are very close to optimal. Moreover, we show that the estimation remains accurate even when a simple empirical estimator of the covariance matrix is used to estimate these directions. These theoretical and numerical results open the way for different generalizations, in particular the incorporation of such ideas in adaptive importance sampling schemes.