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An iterative algorithm for third-order tensor multi-rank minimization

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  • Lei Yang
  • Zheng-Hai Huang
  • Shenglong Hu
  • Jiye Han

Abstract

Recent work by Kilmer et al. (A third-order generalization of the matrix SVD as a product of third-order tensors, Department of Computer Science, Tufts University, Medford, MA, 2008 ; Linear Algebra Appl 435(3):641–658, 2011 ; SIAM J Matrix Anal Appl 34(1):148–172, 2013 ), and Braman (Linear Algebra Appl 433(7):1241–1253, 2010 ) on tensor–tensor multiplication opens up a new avenue to study third-order tensors. Based on this new tensor–tensor multiplication and related concepts, some familiar tools of linear algebra can be extended to study third-order tensors. Motivated by this process, in this paper, we consider the multi-rank of a tensor as a sparsity measure and propose a new model, called third-order tensor multi-rank minimization, as an extension of matrix rank minimization. The operator splitting technique and the convex relaxation technique are used to tackle this problem. Based on these two powerful techniques, we propose a simple first-order and easy-to-implement algorithm to solve this problem. The proposed algorithm is shown to be globally convergent under some assumptions. The continuation technique is also applied to improve the numerical performance of the algorithm. Some preliminary numerical results demonstrate the efficiency of the proposed algorithm, and the potential value and applications of the multi-rank and the tensor multi-rank minimization model. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • Lei Yang & Zheng-Hai Huang & Shenglong Hu & Jiye Han, 2016. "An iterative algorithm for third-order tensor multi-rank minimization," Computational Optimization and Applications, Springer, vol. 63(1), pages 169-202, January.
    • Handle: RePEc:spr:coopap:v:63:y:2016:i:1:p:169-202
    DOI: 10.1007/s10589-015-9769-x
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    References listed on IDEAS

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    1. P. Tseng, 2001. "Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization," Journal of Optimization Theory and Applications, Springer, vol. 109(3), pages 475-494, June.
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    Cited by:

    1. Meng-Meng Zheng & Zheng-Hai Huang & Yong Wang, 2021. "T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming," Computational Optimization and Applications, Springer, vol. 78(1), pages 239-272, January.
    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.

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