Majorization-minimization Bregman proximal gradient algorithms for NMF with the Kullback--Leibler divergence
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908559400239104 |
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| author | Takahashi, Shota Tanaka, Mirai Ikeda, Shiro |
| author_facet | Takahashi, Shota Tanaka, Mirai Ikeda, Shiro |
| contents | Nonnegative matrix factorization (NMF) is a popular method in machine learning and signal processing to decompose a given nonnegative matrix into two nonnegative matrices. In this paper, we propose new algorithms, called majorization-minimization Bregman proximal gradient algorithm (MMBPG) and MMBPG with extrapolation (MMBPGe) to solve NMF. These iterative algorithms minimize the objective function and its potential function monotonically. Assuming the Kurdyka--Łojasiewicz property, we establish that a sequence generated by MMBPG(e) globally converges to a stationary point. We apply MMBPG and MMBPGe to the Kullback--Leibler (KL) divergence-based NMF. While most existing KL-based NMF methods update two blocks or each variable alternately, our algorithms update all variables simultaneously. MMBPG and MMBPGe for KL-based NMF are equipped with a separable Bregman distance that satisfies the smooth adaptable property and that makes its subproblem solvable in closed form. Using this fact, we guarantee that a sequence generated by MMBPG(e) globally converges to a Karush--Kuhn--Tucker (KKT) point of KL-based NMF. In numerical experiments, we compare proposed algorithms with existing algorithms on synthetic data and real-world data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11185 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Majorization-minimization Bregman proximal gradient algorithms for NMF with the Kullback--Leibler divergence Takahashi, Shota Tanaka, Mirai Ikeda, Shiro Optimization and Control 90C26, 49M37, 15A23 Nonnegative matrix factorization (NMF) is a popular method in machine learning and signal processing to decompose a given nonnegative matrix into two nonnegative matrices. In this paper, we propose new algorithms, called majorization-minimization Bregman proximal gradient algorithm (MMBPG) and MMBPG with extrapolation (MMBPGe) to solve NMF. These iterative algorithms minimize the objective function and its potential function monotonically. Assuming the Kurdyka--Łojasiewicz property, we establish that a sequence generated by MMBPG(e) globally converges to a stationary point. We apply MMBPG and MMBPGe to the Kullback--Leibler (KL) divergence-based NMF. While most existing KL-based NMF methods update two blocks or each variable alternately, our algorithms update all variables simultaneously. MMBPG and MMBPGe for KL-based NMF are equipped with a separable Bregman distance that satisfies the smooth adaptable property and that makes its subproblem solvable in closed form. Using this fact, we guarantee that a sequence generated by MMBPG(e) globally converges to a Karush--Kuhn--Tucker (KKT) point of KL-based NMF. In numerical experiments, we compare proposed algorithms with existing algorithms on synthetic data and real-world data. |
| title | Majorization-minimization Bregman proximal gradient algorithms for NMF with the Kullback--Leibler divergence |
| topic | Optimization and Control 90C26, 49M37, 15A23 |
| url | https://arxiv.org/abs/2405.11185 |