A Multilevel Approach to Stochastic Trace Estimation

TL;DR

This paper introduces a multilevel stochastic trace estimation method combining Hutchinson's approach, Chebyshev approximation, and Monte Carlo techniques to efficiently approximate traces of functions of large matrices, reducing variance.

Contribution

It presents a novel multilevel approach that extends existing trace estimation methods, providing theoretical error bounds and demonstrating significant variance reduction in practical applications.

Findings

Multilevel techniques substantially reduce estimator variance.

The method effectively estimates log-determinant, nuclear norm, and Estrada index.

Numerical experiments confirm improved efficiency over single-level estimators.

Abstract

This article presents a randomized matrix-free method for approximating the trace of $f({\bf A})$, where ${\bf A}$ is a large symmetric matrix and $f$ is a function analytic in a closed interval containing the eigenvalues of ${\bf A}$. Our method uses a combination of stochastic trace estimation (i.e., Hutchinson's method), Chebyshev approximation, and multilevel Monte Carlo techniques. We establish general bounds on the approximation error of this method by extending an existing error bound for Hutchinson's method to multilevel trace estimators. Numerical experiments are conducted for common applications such as estimating the log-determinant, nuclear norm, and Estrada index, and triangle counting in graphs. We find that using multilevel techniques can substantially reduce the variance of existing single-level estimators.