A DEIM Induced CUR Factorization

TL;DR

This paper introduces a novel CUR matrix factorization method based on DEIM, providing efficient low-rank approximations with error bounds and demonstrating superior performance over leverage score-based methods.

Contribution

The paper develops a DEIM-based CUR factorization approach, including error analysis and scalable algorithms for large matrices, advancing low-rank approximation techniques.

Findings

Error bounds depend on submatrix conditioning

DEIM-CUR outperforms leverage score methods in experiments

Scalable algorithms for large-scale matrices

Abstract

We derive a CUR matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix $A$, such a factorization provides a low rank approximate decomposition of the form $A \approx C U R$, where $C$ and $R$ are subsets of the columns and rows of $A$, and $U$ is constructed to make $CUR$ a good approximation. Given a low-rank singular value decomposition $A \approx V S W^T$, the DEIM procedure uses $V$ and $W$ to select the columns and rows of $A$ that form $C$ and $R$. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of $V$ and $W$. For large-scale problems, $V$ and $W$ can be approximated using an incremental QR algorithm that makes one pass through $A$. Numerical examples illustrate the favorable performance of the DEIM-CUR method, compared to CUR approximations based on leverage scores.