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Perturbation Analysis on T-Eigenvalues of Third-Order Tensors

Author

Listed:
  • Changxin Mo

    (Chongqing Normal University)

  • Weiyang Ding

    (Fudan University
    Shanghai Center for Brain Science and Brain-Inspired Technology
    Ministry of Education
    Fudan University)

  • Yimin Wei

    (Applied Mathematics, Fudan University)

Abstract

This paper concentrates on perturbation theory concerning the tensor T-eigenvalues within the framework of tensor-tensor multiplication. Notably, it serves as a cornerstone for the extension of semidefinite programming into the domain of tensor fields, referred to as T-semidefinite programming. The analytical perturbation analysis delves into the sensitivity of T-eigenvalues for third-order tensors with square frontal slices, marking the first main part of this study. Three classical results from the matrix domain into the tensor domain are extended. Firstly, this paper presents the Gershgorin disc theorem for tensors, demonstrating the confinement of all T-eigenvalues within a union of Gershgorin discs. Afterward, generalizations of the Bauer-Fike theorem are provided, each applicable to different cases involving tensors, including those that are F-diagonalizable and those that are not. Lastly, the Kahan theorem is presented, addressing the perturbation of a Hermite tensor by any tensors. Additionally, the analysis establishes connections between the T-eigenvalue problem and various optimization problems. The second main part of the paper focuses on tensor pseudospectra theory, presenting four equivalent definitions to characterize tensor $$\varepsilon $$ ε -pseudospectra. Accompanied by a thorough analysis of their properties and illustrative visualizations, this section also explores the application of tensor $$\varepsilon $$ ε -pseudospectra in identifying more T-positive definite tensors.

Suggested Citation

  • Changxin Mo & Weiyang Ding & Yimin Wei, 2024. "Perturbation Analysis on T-Eigenvalues of Third-Order Tensors," Journal of Optimization Theory and Applications, Springer, vol. 202(2), pages 668-702, August.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:2:d:10.1007_s10957-024-02444-z
    DOI: 10.1007/s10957-024-02444-z
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    References listed on IDEAS

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    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.
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