Variance Reduction for Matrix Computations with Applications to Gaussian Processes

TL;DR

This paper introduces variance reduction techniques for matrix computations using factorization, significantly improving efficiency in Gaussian process applications.

Contribution

It presents novel variance reduction methods leveraging matrix factorization, including estimators for trace and log-determinant, with substantial efficiency gains.

Findings

Factorized estimators outperform traditional methods in variance reduction.

Numerical experiments show up to 1,000-fold efficiency improvements.

New estimator for log-determinant enhances Gaussian process likelihood computations.

Abstract

In addition to recent developments in computing speed and memory, methodological advances have contributed to significant gains in the performance of stochastic simulation. In this paper, we focus on variance reduction for matrix computations via matrix factorization. We provide insights into existing variance reduction methods for estimating the entries of large matrices. Popular methods do not exploit the reduction in variance that is possible when the matrix is factorized. We show how computing the square root factorization of the matrix can achieve in some important cases arbitrarily better stochastic performance. In addition, we propose a factorized estimator for the trace of a product of matrices and numerically demonstrate that the estimator can be up to 1,000 times more efficient on certain problems of estimating the log-likelihood of a Gaussian process. Additionally, we provide a new estimator of the log-determinant of a positive semi-definite matrix where the log-determinant is treated as a normalizing constant of a probability density.