On the algorithm of best approximation by low rank matrices in the Chebyshev norm

TL;DR

This paper introduces a new algorithm for low-rank matrix approximation in the Chebyshev norm, enabling efficient and accurate approximations for matrices with slowly decreasing singular values, expanding beyond traditional spectral or Frobenius norm methods.

Contribution

The paper presents a novel method for low-rank approximation specifically in the Chebyshev norm, addressing cases where singular values decrease slowly or not at all.

Findings

Effective approximation for matrices with non-decreasing singular values

Outperforms traditional methods in Chebyshev norm scenarios

Provides theoretical insights into approximation quality

Abstract

The low-rank matrix approximation problem is ubiquitous in computational mathematics. Traditionally, this problem is solved in spectral or Frobenius norms, where the accuracy of the approximation is related to the rate of decrease of the singular values of the matrix. However, recent results indicate that this requirement is not necessary for other norms. In this paper, we propose a method for solving the low-rank approximation problem in the Chebyshev norm, which is capable of efficiently constructing accurate approximations for matrices, whose singular values do not decrease or decrease slowly.