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.

翻译：如今，矩阵的低秩逼近已成为科学与工程领域中众多方法的重要组成部分。传统上，低秩逼近主要在酉不变范数下进行研究，但近年来逐元素逼近在文献中也受到了广泛关注。本文提出一种加速交替最小化算法，用于求解Chebyshev范数下的矩阵低秩逼近问题。通过数值实验，我们证明了所提方法在大规模问题上的有效性。此外，我们从理论上研究了交替最小化方法，并引入了秩r的2-路交错点概念。我们证明了秩r的2-路交错点的存在是Chebyshev范数下最优低秩逼近的必要条件，且交替最小化方法的所有极限点均满足该条件。