Randomized GCUR decompositions

TL;DR

This paper introduces a fast randomized algorithm for generalized CUR decompositions that efficiently approximates large matrices and their low-rank structures, combining sampling, L-DEIM, and RSVD techniques with probabilistic error analysis.

Contribution

It presents a novel randomized approach for generalized CUR decompositions, integrating L-DEIM and RSVD for improved efficiency and accuracy in large-scale data processing.

Findings

The algorithms achieve significant speedups in low-rank matrix approximations.

Numerical results demonstrate the effectiveness and accuracy of the proposed methods.

Probabilistic error bounds validate the reliability of the algorithms.

Abstract

By exploiting the random sampling techniques, this paper derives an efficient randomized algorithm for computing a generalized CUR decomposition, which provides low-rank approximations of both matrices simultaneously in terms of some of their rows and columns. For large-scale data sets that are expensive to store and manipulate, a new variant of the discrete empirical interpolation method known as L-DEIM, which needs much lower cost and provides a significant acceleration in practice, is also combined with the random sampling approach to further improve the efficiency of our algorithm. Moreover, adopting the randomized algorithm to implement the truncation process of restricted singular value decomposition (RSVD), combined with the L-DEIM procedure, we propose a fast algorithm for computing an RSVD based CUR decomposition, which provides a coordinated low-rank approximation of the three matrices in a CUR-type format simultaneously and provides advantages over the standard CUR approximation for some applications. We establish detailed probabilistic error analysis for the algorithms and provide numerical results that show the promise of our approaches.