IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v197y2023i1d10.1007_s10957-023-02167-7.html
   My bibliography  Save this article

Uniqueness in Nuclear Norm Minimization: Flatness of the Nuclear Norm Sphere and Simultaneous Polarization

Author

Listed:
  • Tim Hoheisel

    (McGill University)

  • Elliot Paquette

    (McGill University)

Abstract

In this paper, we establish necessary and sufficient conditions for the existence of line segments (or flats) in the sphere of the nuclear norm via the notion of simultaneous polarization and a refined expression for the subdifferential of the nuclear norm. This is then leveraged to provide (point-based) necessary and sufficient conditions for uniqueness of solutions for minimizing the nuclear norm over an affine subspace. We further establish an alternative set of sufficient conditions for uniqueness, based on the interplay of the subdifferential of the nuclear norm and the range of the problem-defining linear operator. Finally, we show how to transfer the uniqueness results for the original problem to a whole class of nuclear norm-regularized minimization problems with a strictly convex fidelity term.

Suggested Citation

  • Tim Hoheisel & Elliot Paquette, 2023. "Uniqueness in Nuclear Norm Minimization: Flatness of the Nuclear Norm Sphere and Simultaneous Polarization," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 252-276, April.
    • Handle: RePEc:spr:joptap:v:197:y:2023:i:1:d:10.1007_s10957-023-02167-7
    DOI: 10.1007/s10957-023-02167-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-023-02167-7
    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/s10957-023-02167-7?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. Jean Charles Gilbert, 2017. "On the Solution Uniqueness Characterization in the L1 Norm and Polyhedral Gauge Recovery," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 70-101, January.
    2. Hui Zhang & Wotao Yin & Lizhi Cheng, 2015. "Necessary and Sufficient Conditions of Solution Uniqueness in 1-Norm Minimization," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 109-122, January.
    3. Jean-Baptiste Hiriart-Urruty & Hai Le, 2013. "A variational approach of the rank function," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 207-240, July.
    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. Yunier Bello-Cruz & Guoyin Li & Tran Thai An Nghia, 2022. "Quadratic Growth Conditions and Uniqueness of Optimal Solution to Lasso," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 167-190, July.
    2. Roger Behling & Douglas S. Gonçalves & Sandra A. Santos, 2019. "Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 1099-1122, December.
    3. Patrick R. Johnstone & Pierre Moulin, 2017. "Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions," Computational Optimization and Applications, Springer, vol. 67(2), pages 259-292, June.
    4. Fan Wu & Wei Bian, 2020. "Accelerated iterative hard thresholding algorithm for $$l_0$$l0 regularized regression problem," Journal of Global Optimization, Springer, vol. 76(4), pages 819-840, April.
    5. Jean Charles Gilbert, 2017. "On the Solution Uniqueness Characterization in the L1 Norm and Polyhedral Gauge Recovery," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 70-101, January.
    6. James Folberth & Stephen Becker, 2020. "Safe Feature Elimination for Non-negativity Constrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 931-952, March.
    7. Liqun Qi & Ziyan Luo & Qing-Wen Wang & Xinzhen Zhang, 2022. "Quaternion Matrix Optimization: Motivation and Analysis," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 621-648, June.
    8. Yun-Bin Zhao & Houyuan Jiang & Zhi-Quan Luo, 2019. "Weak Stability of ℓ 1 -Minimization Methods in Sparse Data Reconstruction," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 173-195, February.
    9. Abdessamad Barbara & Abderrahim Jourani & Samuel Vaiter, 2019. "Maximal Solutions of Sparse Analysis Regularization," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 374-396, February.
    10. Kim Christensen & Mikkel Slot Nielsen & Mark Podolskij, 2021. "High-dimensional estimation of quadratic variation based on penalized realized variance," Papers 2103.03237, arXiv.org.

    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:197:y:2023:i:1:d:10.1007_s10957-023-02167-7. 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.