Low-rank approximation of parameter-dependent matrices via CUR decomposition

TL;DR

This paper introduces AdaCUR, an efficient, rank-adaptive algorithm for low-rank approximation of parameter-dependent matrices using CUR decompositions, with error control and improved computational complexity.

Contribution

The paper presents AdaCUR, a novel algorithm that reuses column and row indices for nearby parameters, offering efficient, error-controlled low-rank approximations for parameter-dependent matrices.

Findings

ADACUR effectively reuses indices for nearby parameters.

FastAdaCUR provides faster approximations with less accuracy.

Both algorithms have favorable complexity compared to existing methods.

Abstract

A low-rank approximation of a parameter-dependent matrix $A(t)$ is an important task in the computational sciences appearing for example in dynamical systems and compression of a series of images. In this work, we introduce AdaCUR, an efficient algorithm for computing a low-rank approximation of parameter-dependent matrices via CUR decompositions. The key idea for this algorithm is that for nearby parameter values, the column and row indices for the CUR decomposition can often be reused. AdaCUR is rank-adaptive, provides error control, and has complexity that compares favorably against existing methods. A faster algorithm which we call FastAdaCUR that prioritizes speed over accuracy is also given, which is rank-adaptive and has complexity which is at most linear in the number of rows or columns, but without error control.