A Restricted SVD type CUR Decomposition for Matrix Triplets

TL;DR

This paper introduces a novel restricted SVD-based CUR decomposition for matrix triplets, enabling low-rank approximations that leverage selected rows and columns, with applications in multi-view data analysis and noise-perturbed data.

Contribution

The paper proposes a new RSVD-CUR factorization method for matrix triplets using DEIM, providing theoretical error bounds and demonstrating advantages over standard CUR in practical applications.

Findings

The RSVD-CUR provides accurate low-rank approximations within a bounded error.

Numerical experiments show improved performance over standard CUR methods.

Applications include multi-view dimension reduction and noise-perturbed data analysis.

Abstract

We present a new restricted SVD-based CUR (RSVD-CUR) factorization for matrix triplets $(A, B, G)$ that aims to extract meaningful information by providing a low-rank approximation of the three matrices using a subset of their rows and columns. The proposed method utilizes the discrete empirical interpolation method (DEIM) to select the subset of rows and columns from the orthogonal and nonsingular matrices obtained through a restricted singular value decomposition of the matrix triplet. We explore the relationships between a DEIM type RSVD-CUR factorization, a DEIM type CUR factorization, and a DEIM type generalized CUR decomposition, and provide an error analysis that establishes the accuracy of the RSVD-CUR decomposition within a factor of the approximation error of the restricted singular value decomposition of the given matrices. The RSVD-CUR factorization can be used in applications that require approximating one data matrix relative to two other given matrices. We discuss two such applications, namely multi-view dimension reduction and data perturbation problems where a correlated noise matrix is added to the input data matrix. Our numerical experiments demonstrate the advantages of the proposed method over the standard CUR approximation in these scenarios.