Estimating the Trace of the Matrix Inverse by Interpolating from the Diagonal of an Approximate Inverse

TL;DR

This paper introduces a novel method for estimating the trace of a matrix inverse by interpolating from an approximate inverse's diagonal, improving efficiency over traditional Monte Carlo methods especially when the approximation is good.

Contribution

The paper presents a new approach that exploits pattern correlation between the inverse diagonal and an approximate inverse, offering a faster and potentially standalone trace estimation technique.

Findings

Method achieves accurate trace estimates with fewer samples.

Can serve as a variance reduction technique for Monte Carlo.

Demonstrated effectiveness on real application matrices.

Abstract

A number of applications require the computation of the trace of a matrix that is implicitly available through a function. A common example of a function is the inverse of a large, sparse matrix, which is the focus of this paper. When the evaluation of the function is expensive, the task is computationally challenging because the standard approach is based on a Monte Carlo method which converges slowly. We present a different approach that exploits the pattern correlation, if present, between the diagonal of the inverse of the matrix and the diagonal of some approximate inverse that can be computed inexpensively. We leverage various sampling and fitting techniques to fit the diagonal of the approximation to the diagonal of the inverse. Depending on the quality of the approximate inverse, our method may serve as a standalone kernel for providing a fast trace estimate with a small number of samples. Furthermore, the method can be used as a variance reduction method for Monte Carlo in some cases. This is decided dynamically by our algorithm. An extensive set of experiments with various technique combinations on several matrices from some real applications demonstrate the potential of our method.