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T-product factorization based method for matrix and tensor completion problems

Author

Listed:
  • Quan Yu

    (Hunan University)

  • Xinzhen Zhang

    (Tianjin University)

Abstract

Low rank matrix and tensor completion problems are to recover the incomplete two and higher order data of low rank structures. The essential problem in the matrix and tensor completion problems is how to improve the efficiency. For a matrix completion problem, we establish a relationship between matrix rank and tensor tubal rank, and reformulate matrix completion problem as a third order tensor completion problem. For the reformulated tensor completion problem, we adopt a two-stage strategy based on tensor factorization algorithm. In this way, a matrix completion problem of big size can be solved via some matrix computations of smaller sizes. For a third order tensor completion problem, to fully exploit the low rank structures, we introduce the double tubal rank which combines the tubal rank of two tensors, original tensor and the reshaped tensor of the mode-3 unfolding matrix of original tensor. Based on this, we propose a reweighted tensor factorization algorithm for third order tensor completion. Extensive numerical experiments demonstrate that the proposed methods outperform state-of-the-art methods in terms of both accuracy and running time.

Suggested Citation

  • Quan Yu & Xinzhen Zhang, 2023. "T-product factorization based method for matrix and tensor completion problems," Computational Optimization and Applications, Springer, vol. 84(3), pages 761-788, April.
    • Handle: RePEc:spr:coopap:v:84:y:2023:i:3:d:10.1007_s10589-022-00439-y
    DOI: 10.1007/s10589-022-00439-y
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    References listed on IDEAS

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    1. Yu-Fan Li & Kun Shang & Zheng-Hai Huang, 2019. "A singular value p-shrinkage thresholding algorithm for low rank matrix recovery," Computational Optimization and Applications, Springer, vol. 73(2), pages 453-476, June.
    2. Chen Ling & Gaohang Yu & Liqun Qi & Yanwei Xu, 2021. "T-product factorization method for internet traffic data completion with spatio-temporal regularization," Computational Optimization and Applications, Springer, vol. 80(3), pages 883-913, December.
    3. Ledyard Tucker, 1966. "Some mathematical notes on three-mode factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 31(3), pages 279-311, September.
    4. Quan Yu & Xinzhen Zhang, 2022. "A smoothing proximal gradient algorithm for matrix rank minimization problem," Computational Optimization and Applications, Springer, vol. 81(2), pages 519-538, March.
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    Cited by:

    1. Shi-Wei Wang & Guang-Xin Huang & Feng Yin, 2024. "Tensor Conjugate Gradient Methods with Automatically Determination of Regularization Parameters for Ill-Posed Problems with t-Product," Mathematics, MDPI, vol. 12(1), pages 1-20, January.

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