The Rank-1 Completion Problem for Cubic Tensors
<|reference_start|>The Rank-1 Completion Problem for Cubic Tensors: This paper studies the rank-$1$ tensor completion problem for cubic order tensors. First of all, we show that this problem is equivalent to a special rank-$1$ matrix recovery problem. We propose both nuclear norm relaxation and moment relaxation methods for solving the resulting rank-$1$ matrix recovery problem. The nuclear norm relaxation sometimes get 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. In particular, 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. Therefore, much larger size problems can be solved efficiently for strongly rank-$1$ completable tensors. Numerical experiments are shown to demonstrate the efficiency of these proposed methods.<|reference_end|>
@article{zhou2024the,
    title={The Rank-1 Completion Problem for Cubic Tensors},
    author={Jinling Zhou, Jiawang Nie, Zheng Peng, Guangming Zhou},
    journal={arXiv preprint arXiv:2404.08171},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.08171},
    primaryClass={math.OC cs.NA math.NA}
}