FlexTrace: Exchangeable Randomized Trace Estimation for Matrix Functions

TL;DR

FlexTrace is a new single-pass, exchangeable trace estimator for matrix functions that uses only matrix-vector products with the original matrix, offering improved accuracy and efficiency for large symmetric positive semi-definite matrices.

Contribution

We introduce FlexTrace, a novel trace estimation method that requires only matrix-vector products with the original matrix, reducing computational cost and improving accuracy for large matrices.

Findings

FlexTrace outperforms existing trace estimation methods in accuracy.

Theoretical bounds demonstrate FlexTrace's advantages for operator monotone functions.

Numerical experiments confirm FlexTrace's effectiveness in practical applications.

Abstract

We consider the task of estimating the trace of a matrix function, ${\rm tr}(f({\bf A}))$, of a large symmetric positive semi-definite matrix ${\bf A}$. This problem arises in multiple applications, including kernel methods and inverse problems. A key challenge across existing trace estimation methods is the need for matrix-vector products (matvecs) with $f({\bf A})$, which can be very expensive. In this article, we introduce a novel trace estimator, FlexTrace, an exchangeable, single-pass method that estimates ${\rm tr}(f({\bf A}))$ solely using matvecs with ${\bf A}$. We consider the case where $f$ is an operator monotone matrix function with $f(0)=0$, which includes functions such as $\log(1+x)$ and $x^{1/2}$, and derive probabilistic bounds showcasing the theoretical advantages of FlexTrace. Numerical experiments across synthetic examples and application domains demonstrate that FlexTrace provides substantially more accurate estimates of the trace of $f({\bf A})$ compared to existing methods.