A DEIM-CUR factorization with iterative SVDs

TL;DR

This paper introduces an iterative DEIM-CUR factorization method that enhances matrix approximation quality by iteratively selecting columns and rows based on previous selections, outperforming traditional one-round sampling.

Contribution

It develops new iterative subselection strategies based on iterative SVDs to improve DEIM-based CUR factorizations for better matrix approximation.

Findings

Iterative subselection improves approximation accuracy over one-round sampling.

Numerical experiments demonstrate the effectiveness of the proposed iterative strategies.

The method better preserves properties like nonnegativity and sparsity in data matrices.

Abstract

A CUR factorization is often utilized as a substitute for the singular value decomposition (SVD), especially when a concrete interpretation of the singular vectors is challenging. Moreover, if the original data matrix possesses properties like nonnegativity and sparsity, a CUR decomposition can better preserve them compared to the SVD. An essential aspect of this approach is the methodology used for selecting a subset of columns and rows from the original matrix. This study investigates the effectiveness of \emph{one-round sampling} and iterative subselection techniques and introduces new iterative subselection strategies based on iterative SVDs. One provably appropriate technique for index selection in constructing a CUR factorization is the discrete empirical interpolation method (DEIM). Our contribution aims to improve the approximation quality of the DEIM scheme by iteratively invoking it in several rounds, in the sense that we select subsequent columns and rows based on the previously selected ones. Thus, we modify $A$ after each iteration by removing the information that has been captured by the previously selected columns and rows. We also discuss how iterative procedures for computing a few singular vectors of large data matrices can be integrated with the new iterative subselection strategies. We present the results of numerical experiments, providing a comparison of one-round sampling and iterative subselection techniques, and demonstrating the improved approximation quality associated with using the latter.