Randomized Matrix-free Trace and Log-Determinant Estimators

TL;DR

This paper introduces randomized, matrix-free algorithms for efficiently estimating the trace and determinant of Hermitian positive semi-definite matrices, with theoretical error bounds and practical applications demonstrated.

Contribution

It develops new randomized algorithms based on subspace iteration that only require matrix-vector products, providing non-asymptotic error bounds and asymptotic sample complexity analysis.

Findings

Algorithms perform well on low-dimensional matrices

Error bounds are tight and non-asymptotic

Effective in uncertainty quantification applications

Abstract

We present randomized algorithms for estimating the trace and deter- minant of Hermitian positive semi-definite matrices. The algorithms are based on subspace iteration, and access the matrix only through matrix vector products. We analyse the error due to randomization, for starting guesses whose elements are Gaussian or Rademacher random variables. The analysis is cleanly separated into a structural (deterministic) part followed by a probabilistic part. Our absolute bounds for the expectation and concentration of the estimators are non-asymptotic and informative even for matrices of low dimension. For the trace estimators, we also present asymptotic bounds on the number of samples (columns of the starting guess) required to achieve a user-specified relative error. Numerical experiments illustrate the performance of the estimators and the tightness of the bounds on low-dimensional matrices; and on a challenging application in uncertainty quantification arising from Bayesian optimal experimental design.