On the optimal rank-1 approximation of matrices in the Chebyshev norm

TL;DR

This paper explores the problem of finding optimal rank-1 matrix approximations in the Chebyshev norm, proposing an algorithm that effectively constructs such approximations for small matrices, addressing a less-studied norm in low-rank approximation.

Contribution

It introduces an algorithm for optimal rank-1 approximation in Chebyshev norm and analyzes the properties of alternating minimization for this purpose.

Findings

The proposed algorithm successfully constructs optimal rank-1 approximations in Chebyshev norm.

Alternating minimization converges to the optimal solution for small matrices.

The method extends low-rank approximation techniques to a less common norm, broadening application potential.

Abstract

The problem of low rank approximation is ubiquitous in science. Traditionally this problem is solved in unitary invariant norms such as Frobenius or spectral norm due to existence of efficient methods for building approximations. However, recent results reveal the potential of low rank approximations in Chebyshev norm, which naturally arises in many applications. In this paper we tackle the problem of building optimal rank-1 approximations in the Chebyshev norm. We investigate the properties of alternating minimization algorithm for building the low rank approximations and demonstrate how to use it to construct optimal rank-1 approximation. As a result we propose an algorithm that is capable of building optimal rank-1 approximations in Chebyshev norm for small matrices.