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Regularized and Structured Tensor Total Least Squares Methods with Applications

Author

Listed:
  • Feiyang Han

    (Fudan University)

  • Yimin Wei

    (Fudan University)

  • Pengpeng Xie

    (Ocean University of China)

Abstract

Total least squares (TLS), also named as errors in variables in statistical analysis, is an effective method for solving linear equations with the situations, when noise is not just in observation data but also in mapping operations. Besides, the Tikhonov regularization is widely considered in plenty of ill-posed problems. Moreover, the structure of mapping operator plays a crucial role in solving the TLS problem. Tensor operators have some advantages over the characterization of models, which requires us to build the corresponding theory on the tensor TLS. This paper proposes tensor regularized TLS and structured tensor TLS methods for solving ill-conditioned and structured tensor equations, respectively, adopting a tensor-tensor-product. Properties and algorithms for the solution of these approaches are also presented and proved. Based on this method, some applications in image and video deblurring are explored. Numerical examples illustrate the effectiveness of our methods, compared with some existing methods.

Suggested Citation

  • Feiyang Han & Yimin Wei & Pengpeng Xie, 2024. "Regularized and Structured Tensor Total Least Squares Methods with Applications," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1101-1136, September.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:3:d:10.1007_s10957-024-02507-1
    DOI: 10.1007/s10957-024-02507-1
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    References listed on IDEAS

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    1. A. H. Bentbib & A. El Hachimi & K. Jbilou & A. Ratnani, 2022. "A Tensor Regularized Nuclear Norm Method for Image and Video Completion," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 401-425, February.
    2. Hua Zhou & Lexin Li & Hongtu Zhu, 2013. "Tensor Regression with Applications in Neuroimaging Data Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(502), pages 540-552, June.
    3. Xuezhong Wang & Maolin Che & Yimin Wei, 2020. "Tensor neural network models for tensor singular value decompositions," Computational Optimization and Applications, Springer, vol. 75(3), pages 753-777, April.
    4. Xuezhong Wang & Ping Wei & Yimin Wei, 2023. "A Fixed Point Iterative Method for Third-order Tensor Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 334-357, April.
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