Variance Reduction for Inverse Trace Estimation via Random Spanning Forests

TL;DR

This paper improves the variance reduction of forest-based estimators for inverse trace computation in large matrices, making them more efficient and competitive with existing methods.

Contribution

It introduces simple, effective variance reduction techniques—control variates and stratified sampling—for forest-based estimators in inverse trace estimation.

Findings

Significant variance reduction achieved

Comparable or improved performance over state-of-the-art methods

Easy implementation of proposed techniques

Abstract

The trace $\tr(q(\ma{L} + q\ma{I})^{-1})$, where $\ma{L}$ is a symmetric diagonally dominant matrix, is the quantity of interest in some machine learning problems. However, its direct computation is impractical if the matrix size is large. State-of-the-art methods include Hutchinson's estimator combined with iterative solvers, as well as the estimator based on random spanning forests (a random process on graphs). In this work, we show two ways of improving the forest-based estimator via well-known variance reduction techniques, namely control variates and stratified sampling. Implementing these techniques is easy, and provides substantial variance reduction, yielding comparable or better performance relative to state-of-the-art algorithms.