2024/10/07 by Morozov, Stanislav, Zheltkov, Dmitry, Osinsky, Alexander · 1 citation
#15A23 #41A50 #65F55 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2410.05247
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.