Stochastic determination of matrix determinants

TL;DR

This paper introduces a stochastic probing method to estimate the logarithm of a matrix determinant using only matrix-vector multiplications, facilitating large-scale Bayesian inference tasks.

Contribution

A novel stochastic approach for estimating matrix determinants from routine matrix-vector operations, enabling applications in large-scale data analysis and Bayesian inference.

Findings

Effective stochastic estimator for log-determinant introduced

Applicable to large matrices where direct computation is infeasible

Enhances Bayesian model comparison and evidence calculation

Abstract

Matrix determinants play an important role in data analysis, in particular when Gaussian processes are involved. Due to currently exploding data volumes, linear operations - matrices - acting on the data are often not accessible directly but are only represented indirectly in form of a computer routine. Such a routine implements the transformation a data vector undergoes under matrix multiplication. While efficient probing routines to estimate a matrix's diagonal or trace, based solely on such computationally affordable matrix-vector multiplications, are well known and frequently used in signal inference, there is no stochastic estimate for its determinant. We introduce a probing method for the logarithm of a determinant of a linear operator. Our method rests upon a reformulation of the log-determinant by an integral representation and the transformation of the involved terms into stochastic expressions. This stochastic determinant determination enables large-size applications in Bayesian inference, in particular evidence calculations, model comparison, and posterior determination.