Faster Stochastic Trace Estimation with a Chebyshev Product Identity

TL;DR

This paper introduces a more efficient method for stochastic trace estimation that reduces the number of matrix-vector products needed by leveraging a Chebyshev product identity, significantly speeding up computations.

Contribution

It presents a novel approach to evaluate polynomial expressions in symmetric matrices with fewer matrix-vector products, improving the efficiency of trace estimation algorithms.

Findings

Reduces matrix-vector products from n to approximately n/2

Speeds up stochastic trace estimation methods

Applicable to Chebyshev and Taylor series-based algorithms

Abstract

Methods for stochastic trace estimation often require the repeated evaluation of expressions of the form $z^T p_n(A)z$, where $A$ is a symmetric matrix and $p_n$ is a degree $n$ polynomial written in the standard or Chebyshev basis. We show how to evaluate these expressions using only $\lceil n/2\rceil$ matrix-vector products, thus substantially reducing the cost of existing trace estimation algorithms that use Chebyshev interpolation or Taylor series.