FlexTrace: Exchangeable Randomized Trace Estimation for Matrix Functions

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.

💡 Research Summary

This paper addresses the problem of estimating the trace of a matrix function, tr f(A), for a large symmetric positive‑semidefinite (SPSD) matrix A. Such quantities appear in many areas, including Gaussian‑process log‑determinants, Schatten‑norms, and kernel‑based effective dimensionality. Existing stochastic trace estimators either require matrix‑vector products with the function‑applied matrix f(A) (e.g., Stochastic Lanczos Quadrature) or need multiple passes over A, which is prohibitive when each matvec with A is expensive or only available offline.

The authors propose FlexTrace, a single‑pass, exchangeable estimator that uses only matvecs with A. The method proceeds in two stages. First, a random Gaussian test matrix Ω∈ℝ^{n×k} is drawn and a randomized Nyström approximation of A is built: \