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arXiv · 2410.05247

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

Stanislav Morozov·Dmitry Zheltkov·Alexander Osinsky
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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.

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Stanislav Morozov, Dmitry Zheltkov, Alexander Osinsky. 2024-10-07. Accelerated alternating minimization algorithm for low-rank approximations in the Chebyshev norm. https://arxiv.org/abs/2410.05247

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