Low-rank updates of matrix functions

TL;DR

This paper introduces an efficient method for updating matrix functions after low-rank modifications, leveraging tensorized Krylov subspaces, with proven convergence and applications in network analysis.

Contribution

It proposes a novel Krylov subspace-based approach for low-rank updates of matrix functions, with theoretical convergence guarantees and practical efficiency.

Findings

Method is exact for polynomial functions up to degree m.

Convergence results are established for exponential and Markov functions.

Effective in updating network centrality and communicability measures.

Abstract

We consider the task of updating a matrix function $f(A)$ when the matrix $A\in{\mathbb C}^{n \times n}$ is subject to a low-rank modification. In other words, we aim at approximating $f(A+D)-f(A)$ for a matrix $D$ of rank $k \ll n$. The approach proposed in this paper attains efficiency by projecting onto tensorized Krylov subspaces produced by matrix-vector multiplications with $A$ and $A^*$. We prove the approximations obtained from $m$ steps of the proposed methods are exact if $f$ is a polynomial of degree at most $m$ and use this as a basis for proving a variety of convergence results, in particular for the matrix exponential and for Markov functions. We illustrate the performance of our method by considering various examples from network analysis, where our approach can be used to cheaply update centrality and communicability measures.