The Rank-1 Completion Problem for Cubic Tensors
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929552892100608 |
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| author | Zhou, Jinling Nie, Jiawang Peng, Zheng Zhou, Guangming |
| author_facet | Zhou, Jinling Nie, Jiawang Peng, Zheng Zhou, Guangming |
| contents | This paper studies the rank-$1$ tensor completion problem for cubic tensors. First of all, we show that this problem is equivalent to a special rank-$1$ matrix recovery problem. When the tensor is strongly rank-$1$ completable, we show that the problem is equivalent to a rank-$1$ matrix completion problem and it can be solved by an iterative formula. For other cases, we propose both nuclear norm relaxation and moment relaxation methods for solving the resulting rank-$1$ matrix recovery problem. The nuclear norm relaxation sometimes returns a rank-$1$ tensor completion, while sometimes it does not. When it fails, we apply the moment hierarchy of semidefinite programming relaxations to solve the rank-$1$ matrix recovery problem. The moment hierarchy can always get a rank-$1$ tensor completion, or detect its nonexistence. Numerical experiments are shown to demonstrate the efficiency of these proposed methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08171 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Rank-1 Completion Problem for Cubic Tensors Zhou, Jinling Nie, Jiawang Peng, Zheng Zhou, Guangming Optimization and Control Numerical Analysis This paper studies the rank-$1$ tensor completion problem for cubic tensors. First of all, we show that this problem is equivalent to a special rank-$1$ matrix recovery problem. When the tensor is strongly rank-$1$ completable, we show that the problem is equivalent to a rank-$1$ matrix completion problem and it can be solved by an iterative formula. For other cases, we propose both nuclear norm relaxation and moment relaxation methods for solving the resulting rank-$1$ matrix recovery problem. The nuclear norm relaxation sometimes returns a rank-$1$ tensor completion, while sometimes it does not. When it fails, we apply the moment hierarchy of semidefinite programming relaxations to solve the rank-$1$ matrix recovery problem. The moment hierarchy can always get a rank-$1$ tensor completion, or detect its nonexistence. Numerical experiments are shown to demonstrate the efficiency of these proposed methods. |
| title | The Rank-1 Completion Problem for Cubic Tensors |
| topic | Optimization and Control Numerical Analysis |
| url | https://arxiv.org/abs/2404.08171 |