Perturbations of CUR Decompositions

TL;DR

This paper analyzes how noise impacts CUR matrix decompositions, providing perturbation estimates for various variants and demonstrating how column and row choices influence approximation quality.

Contribution

It offers a comprehensive perturbation analysis of CUR decompositions under noise, including new bounds and insights into the effects of column and row selection.

Findings

Perturbation estimates depend on noise magnitude and matrix norms.

Choice of columns and rows significantly affects approximation quality.

New state-of-the-art bounds for certain CUR variants.

Abstract

The CUR decomposition is a factorization of a low-rank matrix obtained by selecting certain column and row submatrices of it. We perform a thorough investigation of what happens to such decompositions in the presence of noise. Since CUR decompositions are non-uniquely formed, we investigate several variants and give perturbation estimates for each in terms of the magnitude of the noise matrix in a broad class of norms which includes all Schatten $p$--norms. The estimates given here are qualitative and illustrate how the choice of columns and rows affects the quality of the approximation, and additionally we obtain new state-of-the-art bounds for some variants of CUR approximations.