A Randomized Algorithm for Approximating the Log Determinant of a Symmetric Positive Definite Matrix

TL;DR

This paper presents a new randomized algorithm that efficiently approximates the log determinant of large symmetric positive definite matrices with theoretical error bounds and practical accuracy demonstrated through implementation.

Contribution

It introduces a novel randomized method with theoretical guarantees for approximating the log determinant of SPD matrices, improving efficiency and accuracy.

Findings

The algorithm provides accurate approximations in seconds for large matrices.

It offers additive and relative error bounds applicable to different classes of SPD matrices.

Empirical results show high accuracy of the approximation in practical scenarios.

Abstract

We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (SPD) matrix. The algorithm is randomized and approximates the traces of a small number of matrix powers of a specially constructed matrix, using the method of Avron and Toledo~\cite{AT11}. From a theoretical perspective, we present additive and relative error bounds for our algorithm. Our additive error bound works for any SPD matrix, whereas our relative error bound works for SPD matrices whose eigenvalues lie in the interval $(\theta_1,1)$, with $0<\theta_1<1$; the latter setting was proposed in~\cite{icml2015_hana15}. From an empirical perspective, we demonstrate that a C++ implementation of our algorithm can approximate the logarithm of the determinant of large matrices very accurately in a matter of seconds.