IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v63y2016i1p169-202.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-015-9769-x
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1007/s10589-015-9769-x?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. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    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. Shuang Zhang & Xingdong Feng, 2022. "Distributed identification of heterogeneous treatment effects," Computational Statistics, Springer, vol. 37(1), pages 57-89, March.
    2. Jung, Yoon Mo & Whang, Joyce Jiyoung & Yun, Sangwoon, 2020. "Sparse probabilistic K-means," Applied Mathematics and Computation, Elsevier, vol. 382(C).
    3. Seunghwan Lee & Sang Cheol Kim & Donghyeon Yu, 2023. "An efficient GPU-parallel coordinate descent algorithm for sparse precision matrix estimation via scaled lasso," Computational Statistics, Springer, vol. 38(1), pages 217-242, March.
    4. Le Thi Khanh Hien & Duy Nhat Phan & Nicolas Gillis, 2022. "Inertial alternating direction method of multipliers for non-convex non-smooth optimization," Computational Optimization and Applications, Springer, vol. 83(1), pages 247-285, September.
    5. Victor Chernozhukov & Whitney K. Newey & Victor Quintas-Martinez & Vasilis Syrgkanis, 2021. "Automatic Debiased Machine Learning via Riesz Regression," Papers 2104.14737, arXiv.org, revised Mar 2024.
    6. Paul Tseng, 2004. "An Analysis of the EM Algorithm and Entropy-Like Proximal Point Methods," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 27-44, February.
    7. Omer, E. & Guetta, R. & Ioslovich, I. & Gutman, P.O. & Borshchevsky, M., 2008. "“Energy Tower” combined with pumped storage and desalination: Optimal design and analysis," Renewable Energy, Elsevier, vol. 33(4), pages 597-607.
    8. David Degras, 2021. "Sparse group fused lasso for model segmentation: a hybrid approach," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 15(3), pages 625-671, September.
    9. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "A Block Inertial Bregman Proximal Algorithm for Nonsmooth Nonconvex Problems with Application to Symmetric Nonnegative Matrix Tri-Factorization," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 234-258, July.
    10. Astorino, Annabella & Avolio, Matteo & Fuduli, Antonio, 2022. "A maximum-margin multisphere approach for binary Multiple Instance Learning," European Journal of Operational Research, Elsevier, vol. 299(2), pages 642-652.
    11. Fang, Kuangnan & Wang, Xiaoyan & Shia, Ben-Chang & Ma, Shuangge, 2016. "Identification of proportionality structure with two-part models using penalization," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 12-24.
    12. Z. John Daye & Jinbo Chen & Hongzhe Li, 2012. "High-Dimensional Heteroscedastic Regression with an Application to eQTL Data Analysis," Biometrics, The International Biometric Society, vol. 68(1), pages 316-326, March.
    13. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2015. "Optimal Replenishment Order Placement in a Finite Time Horizon," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 1078-1089, March.
    14. Griveau-Billion, Théophile & Richard, Jean-Charles & Roncalli, Thierry, 2013. "A Fast Algorithm for Computing High-dimensional Risk Parity Portfolios," MPRA Paper 49822, University Library of Munich, Germany.
    15. Fang, Kuangnan & Chen, Yuanxing & Ma, Shuangge & Zhang, Qingzhao, 2022. "Biclustering analysis of functionals via penalized fusion," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    16. Yanming Li & Bin Nan & Ji Zhu, 2015. "Multivariate sparse group lasso for the multivariate multiple linear regression with an arbitrary group structure," Biometrics, The International Biometric Society, vol. 71(2), pages 354-363, June.
    17. P. Tseng & S. Yun, 2009. "Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 513-535, March.
    18. Murat Genç, 2022. "A new double-regularized regression using Liu and lasso regularization," Computational Statistics, Springer, vol. 37(1), pages 159-227, March.
    19. Justin A. Sirignano & Gerry Tsoukalas & Kay Giesecke, 2016. "Large-Scale Loan Portfolio Selection," Operations Research, INFORMS, vol. 64(6), pages 1239-1255, December.
    20. Sven Jäger & Anita Schöbel, 2020. "The blockwise coordinate descent method for integer programs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(2), pages 357-381, 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:coopap:v:63:y:2016:i:1:p:169-202. 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.