Accelerated alternating minimization algorithm for low-rank approximations in the Chebyshev norm
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| Format: | Preprint |
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2024
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| _version_ | 1866910219591745536 |
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| author | Morozov, Stanislav Zheltkov, Dmitry Osinsky, Alexander |
| author_facet | Morozov, Stanislav Zheltkov, Dmitry Osinsky, Alexander |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_05247 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Accelerated alternating minimization algorithm for low-rank approximations in the Chebyshev norm Morozov, Stanislav Zheltkov, Dmitry Osinsky, Alexander Numerical Analysis 15A23, 65F55, 41A50 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. |
| title | Accelerated alternating minimization algorithm for low-rank approximations in the Chebyshev norm |
| topic | Numerical Analysis 15A23, 65F55, 41A50 |
| url | https://arxiv.org/abs/2410.05247 |