Estimating the inverse trace using random forests on graphs

TL;DR

This paper introduces an unbiased estimator for the inverse trace of large matrices using Wilson's algorithm, leveraging graph-based sampling, which is efficient and scalable but limited to diagonally dominant matrices.

Contribution

The paper presents a novel unbiased estimator for regularized inverse traces based on Wilson's algorithm, extending trace estimation techniques to large, diagonally dominant matrices.

Findings

Method is fast and easy to implement.

Scales efficiently to very large matrices.

Limited to diagonally dominant matrices.

Abstract

Some data analysis problems require the computation of (regularised) inverse traces, i.e. quantities of the form $\Tr (q \bI + \bL)^{-1}$. For large matrices, direct methods are unfeasible and one must resort to approximations, for example using a conjugate gradient solver combined with Girard's trace estimator (also known as Hutchinson's trace estimator). Here we describe an unbiased estimator of the regularized inverse trace, based on Wilson's algorithm, an algorithm that was initially designed to draw uniform spanning trees in graphs. Our method is fast, easy to implement, and scales to very large matrices. Its main drawback is that it is limited to diagonally dominant matrices $\bL$.