Analysis of probing techniques for sparse approximation and trace estimation of decaying matrix functions

TL;DR

This paper provides new theoretical insights and error bounds for probing techniques used in approximating sparse matrix functions and their traces, especially when these functions exhibit decay properties, enhancing computational efficiency.

Contribution

The paper develops rigorous error bounds for probing methods based on graph colorings and offers practical stopping criteria for Krylov subspace approximations.

Findings

Established error bounds for probing methods with decaying matrix functions.

Provided criteria for efficient Krylov iteration stopping.

Enhanced understanding of when sparse approximations are effective.

Abstract

The computation of matrix functions $f(A)$, or related quantities like their trace, is an important but challenging task, in particular for large and sparse matrices $A$. In recent years, probing methods have become an often considered tool in this context, as they allow to replace the computation of $f(A)$ or $\text{tr}(f(A))$ by the evaluation of (a small number of) quantities of the form $f(A)v$ or $v^Tf(A)v$, respectively. These tasks can then efficiently be solved by standard techniques like, e.g., Krylov subspace methods. It is well-known that probing methods are particularly efficient when $f(A)$ is approximately sparse, e.g., when the entries of $f(A)$ show a strong off-diagonal decay, but a rigorous error analysis is lacking so far. In this paper we develop new theoretical results on the existence of sparse approximations for $f(A)$ and error bounds for probing methods based on graph colorings. As a by-product, by carefully inspecting the proofs of these error bounds, we also gain new insights into when to stop the Krylov iteration used for approximating $f(A)v$ or $v^Tf(A)v$, thus allowing for a practically efficient implementation of the probing methods.