IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v75y2020i3d10.1007_s10589-020-00177-z.html
   My bibliography  Save this article

On the tensor spectral p-norm and its dual norm via partitions

Author

Listed:
  • Bilian Chen

    (Xiamen University
    Xiamen Key Laboratory of Big Data Intelligent Analysis and Decision)

  • Zhening Li

    (University of Portsmouth)

Abstract

This paper presents a generalization of the spectral norm and the nuclear norm of a tensor via arbitrary tensor partitions, a much richer concept than block tensors. We show that the spectral p-norm and the nuclear p-norm of a tensor can be lower and upper bounded by manipulating the spectral p-norms and the nuclear p-norms of subtensors in an arbitrary partition of the tensor for $$1\le p\le \infty$$1≤p≤∞. Hence, it generalizes and answers affirmatively the conjecture proposed by Li (SIAM J Matrix Anal Appl 37:1440–1452, 2016) for a tensor partition and $$p=2$$p=2. We study the relations of the norms of a tensor, the norms of matrix unfoldings of the tensor, and the bounds via the norms of matrix slices of the tensor. Various bounds of the tensor spectral and nuclear norms in the literature are implied by our results.

Suggested Citation

  • Bilian Chen & Zhening Li, 2020. "On the tensor spectral p-norm and its dual norm via partitions," Computational Optimization and Applications, Springer, vol. 75(3), pages 609-628, April.
    • Handle: RePEc:spr:coopap:v:75:y:2020:i:3:d:10.1007_s10589-020-00177-z
    DOI: 10.1007/s10589-020-00177-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-020-00177-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/s10589-020-00177-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. Simai He & Bo Jiang & Zhening Li & Shuzhong Zhang, 2014. "Probability Bounds for Polynomial Functions in Random Variables," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 889-907, August.
    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. Yuning Yang, 2022. "On Approximation Algorithm for Orthogonal Low-Rank Tensor Approximation," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 821-851, September.
    2. Xianpeng Mao & Yuning Yang, 2022. "Several approximation algorithms for sparse best rank-1 approximation to higher-order tensors," Journal of Global Optimization, Springer, vol. 84(1), pages 229-253, September.
    3. Ke Hou & Anthony Man-Cho So, 2014. "Hardness and Approximation Results for L p -Ball Constrained Homogeneous Polynomial Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1084-1108, November.
    4. Taoran Fu & Bo Jiang & Zhening Li, 2018. "Approximation algorithms for optimization of real-valued general conjugate complex forms," Journal of Global Optimization, Springer, vol. 70(1), pages 99-130, January.
    5. Bo Jiang & Zhening Li & Shuzhong Zhang, 2014. "Approximation methods for complex polynomial optimization," Computational Optimization and Applications, Springer, vol. 59(1), pages 219-248, October.

    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:coopap:v:75:y:2020:i:3:d:10.1007_s10589-020-00177-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.