IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v202y2024i2d10.1007_s10957-024-02444-z.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02444-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-024-02444-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hiroki Marumo & Sunyoung Kim & Makoto Yamashita, 2024. "T-semidefinite programming relaxation with third-order tensors for constrained polynomial optimization," Computational Optimization and Applications, Springer, vol. 89(1), pages 183-218, September.
    2. Meng-Meng Zheng & Zheng-Hai Huang & Sheng-Long Hu, 2022. "Unconstrained minimization of block-circulant polynomials via semidefinite program in third-order tensor space," Journal of Global Optimization, Springer, vol. 84(2), pages 415-440, October.
    3. Huang, Baohua, 2024. "Conjugate gradient-type method for the tensor linear system via the T-product and its application in the calculation of Moore-Penrose inverse," Applied Mathematics and Computation, Elsevier, vol. 472(C).
    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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:202:y:2024:i:2:d:10.1007_s10957-024-02444-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.