On randomized trace estimates for indefinite matrices with an application to determinants

TL;DR

This paper develops improved tail bounds for randomized trace estimation of indefinite matrices, enhancing efficiency in applications like log-determinant approximation crucial for statistical learning tasks.

Contribution

It introduces new tail bounds for indefinite matrices using Rademacher or Gaussian vectors, reducing sample complexity and extending analysis to matrix functions like the logarithm.

Findings

Significantly reduces the number of samples needed for trace estimation.

Extends tail bounds to indefinite matrices, not just SPD.

Improves existing bounds for the matrix logarithm approximation.

Abstract

Randomized trace estimation is a popular and well studied technique that approximates the trace of a large-scale matrix $B$ by computing the average of $x^T Bx$ for many samples of a random vector $X$. Often, $B$ is symmetric positive definite (SPD) but a number of applications give rise to indefinite $B$. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite $B$ with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite $B$, reducing the the number of required samples by a factor $n$ or even more, where $n$ is the size of $B$. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of $f(A)$. Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.