# Low-rank updates of matrix functions

arXiv: https://arxiv.org/abs/1707.03045 · HTML: https://mlsanity.com/paper/1707.03045/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jul 2017.

## Built on by (most cited first, 3 of 3)

- [Low-Rank Updates of Matrix Square Roots](https://mlsanity.com/paper/2201.13156/) (2022)
- [Dynamic estimation of slowly varying sequences](https://mlsanity.com/paper/2606.23655/) (2026)
- [Approximating Measures on Function Spaces: Transport and Truncation](https://mlsanity.com/paper/2609.17802/) (2026)

## Abstract (arXiv)

We consider the task of updating a matrix function f(A) when the matrix A∈mathbb Cn × 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.