Detecting when one probe vector is enough for preconditioned log-determinant approximation

TL;DR

This paper introduces randomized algorithms for efficiently estimating the log-determinant of regularized symmetric positive semi-definite matrices, utilizing preconditioning and stochastic trace estimation to reduce computational cost.

Contribution

It proposes a strategy combining Nyström preconditioning and residual estimation with SLQ, along with a detection algorithm to adaptively choose the best approach.

Findings

The proposed method achieves small errors with minimal samples.

The detection algorithm effectively identifies when more samples are needed.

Numerical tests show competitiveness with existing algorithms.

Abstract

We present randomized algorithms for estimating the log-determinant of regularized symmetric positive semi-definite matrices. The algorithms access the matrix only through matrix vector products, and are based on the introduction of a preconditioner and stochastic trace estimator. We claim that preconditioning as much as we can and making a rough estimate of the residual part with a small budget achieves a small error in most of the cases. We choose a Nystr\"om preconditioner and estimate the residual using only one sample of stochastic Lanczos quadrature (SLQ). We analyze the performance of this strategy from a theoretical and practical viewpoint. We also present an algorithm that, at almost no additional cost, detects whether the proposed strategy is not the most effective, in which case it uses more samples for the SLQ part. Numerical examples on several test matrices show that our proposed methods are competitive with existing algorithms.