IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v70y2018i1d10.1007_s10898-017-0573-2.html
   My bibliography  Save this article

Two-stage convex relaxation approach to low-rank and sparsity regularized least squares loss

Author

Listed:
  • Le Han

    (South China University of Technology)

  • Shujun Bi

    (South China University of Technology)

Abstract

In this paper we consider the rank and zero norm regularized least squares loss minimization problem with a spectral norm ball constraint. For this class of NP-hard optimization problems, we propose a two-stage convex relaxation approach by majorizing some suitable locally Lipschitz continuous surrogates. Furthermore, the Frobenius norm error bound for the optimal solution of each stage is characterized and the theoretical guarantee is established for the two-stage convex relaxation approach by showing that the error bound of the first stage convex relaxation (i.e., the nuclear norm and $$\ell _1$$ ℓ 1 -norm regularized minimization problem), can be reduced much by the second stage convex relaxation under a suitable restricted eigenvalue condition. Also, we verify the efficiency of the proposed approach by applying it to some random test problems and some real problems.

Suggested Citation

  • Le Han & Shujun Bi, 2018. "Two-stage convex relaxation approach to low-rank and sparsity regularized least squares loss," Journal of Global Optimization, Springer, vol. 70(1), pages 71-97, January.
    • Handle: RePEc:spr:jglopt:v:70:y:2018:i:1:d:10.1007_s10898-017-0573-2
    DOI: 10.1007/s10898-017-0573-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-017-0573-2
    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/s10898-017-0573-2?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. Le Han & Shujun Bi & Shaohua Pan, 2016. "Two-stage convex relaxation approach to least squares loss constrained low-rank plus sparsity optimization problems," Computational Optimization and Applications, Springer, vol. 64(1), pages 119-148, May.
    2. Necdet Aybat & Donald Goldfarb & Shiqian Ma, 2014. "Efficient algorithms for robust and stable principal component pursuit problems," Computational Optimization and Applications, Springer, vol. 58(1), pages 1-29, May.
    3. Lingchen Kong & Naihua Xiu, 2013. "EXACT LOW-RANK MATRIX RECOVERY VIA NONCONVEX SCHATTEN p-MINIMIZATION," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 30(03), pages 1-13.
    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. Zhang, Shuang & Han, Le, 2023. "Robust tensor recovery with nonconvex and nonsmooth regularization," Applied Mathematics and Computation, Elsevier, vol. 438(C).

    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. Le Han & Shujun Bi & Shaohua Pan, 2016. "Two-stage convex relaxation approach to least squares loss constrained low-rank plus sparsity optimization problems," Computational Optimization and Applications, Springer, vol. 64(1), pages 119-148, May.
    2. Zhaosong Lu & Yong Zhang & Jian Lu, 2017. "$$\ell _p$$ ℓ p Regularized low-rank approximation via iterative reweighted singular value minimization," Computational Optimization and Applications, Springer, vol. 68(3), pages 619-642, December.
    3. N. Aybat & G. Iyengar, 2015. "An alternating direction method with increasing penalty for stable principal component pursuit," Computational Optimization and Applications, Springer, vol. 61(3), pages 635-668, July.
    4. Yu-Fan Li & Kun Shang & Zheng-Hai Huang, 2019. "A singular value p-shrinkage thresholding algorithm for low rank matrix recovery," Computational Optimization and Applications, Springer, vol. 73(2), pages 453-476, June.

    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:jglopt:v:70:y:2018:i:1:d:10.1007_s10898-017-0573-2. 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.