Accelerated alternating minimization algorithm for low-rank approximations in the Chebyshev norm

TL;DR

This paper introduces an accelerated alternating minimization algorithm for low-rank matrix approximation in the Chebyshev norm, demonstrating its effectiveness for large-scale problems and analyzing its theoretical properties.

Contribution

It proposes a novel accelerated algorithm for Chebyshev norm low-rank approximation and establishes theoretical conditions for optimality involving 2-way alternance.

Findings

The algorithm is effective for large-scale matrix problems.

Presence of 2-way alternance is necessary for optimal approximation.

All limit points of the method satisfy the 2-way alternance condition.

Abstract

Nowadays, low-rank approximations of matrices are an important component of many methods in science and engineering. Traditionally, low-rank approximations are considered in unitary invariant norms, however, recently element-wise approximations have also received significant attention in the literature. In this paper, we propose an accelerated alternating minimization algorithm for solving the problem of low-rank approximation of matrices in the Chebyshev norm. Through the numerical evaluation we demonstrate the effectiveness of the proposed procedure for large-scale problems. We also theoretically investigate the alternating minimization method and introduce the notion of a $2$-way alternance of rank $r$. We show that the presence of a $2$-way alternance of rank $r$ is the necessary condition of the optimal low-rank approximation in the Chebyshev norm and that all limit points of the alternating minimization method satisfy this condition.