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Gauss–Southwell Type Descent Methods for Low-Rank Matrix Optimization

Author

Listed:
  • Guillaume Olikier

    (Université Côte d’Azur and Inria)

  • André Uschmajew

    (University of Augsburg)

  • Bart Vandereycken

    (University of Geneva)

Abstract

We consider gradient-related methods for low-rank matrix optimization with a smooth cost function. The methods operate on single factors of the low-rank factorization and share aspects of both alternating and Riemannian optimization. Two possible choices for the search directions based on Gauss–Southwell type selection rules are compared: one using the gradient of a factorized non-convex formulation, the other using the Riemannian gradient. While both methods provide gradient convergence guarantees that are similar to the unconstrained case, numerical experiments on a quadratic cost function indicate that the version based on the Riemannian gradient is significantly more robust with respect to small singular values and the condition number of the cost function. As a side result of our approach, we also obtain new convergence results for the alternating least squares method.

Suggested Citation

  • Guillaume Olikier & André Uschmajew & Bart Vandereycken, 2025. "Gauss–Southwell Type Descent Methods for Low-Rank Matrix Optimization," Journal of Optimization Theory and Applications, Springer, vol. 206(1), pages 1-32, July.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:1:d:10.1007_s10957-025-02682-9
    DOI: 10.1007/s10957-025-02682-9
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    References listed on IDEAS

    as
    1. Yuetian Luo & Xudong Li & Anru R. Zhang, 2024. "On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-Rank Matrix Optimization," Mathematics of Operations Research, INFORMS, vol. 49(2), pages 782-825, May.
    2. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, April.
    3. NESTEROV, Yurii, 2012. "Efficiency of coordinate descent methods on huge-scale optimization problems," LIDAM Reprints CORE 2511, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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