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A singular value shrinkage thresholding algorithm for folded concave penalized low-rank matrix optimization problems

Author

Listed:
  • Xian Zhang

    (Guizhou University)

  • Dingtao Peng

    (Guizhou University)

  • Yanyan Su

    (Guizhou University)

Abstract

In this paper, we study the low-rank matrix optimization problem, where the loss function is smooth but not necessarily convex, and the penalty term is a nonconvex (folded concave) continuous relaxation of the rank function. Firstly, we give the closed-form singular value shrinkage thresholding operators for several matrix-valued folded concave penalty functions. Secondly, we adopt a singular value shrinkage thresholding (SVST) algorithm for the nonconvex low-rank matrix optimization problem, and prove that the proposed SVST algorithm converges to a stationary point of the problem. Furthermore, we show that the limit point satisfies a global necessary optimality condition which can exclude too many stationary points even local minimizers in order to refine the solutions. We conduct a large number of numeric experiments to test the performance of SVST algorithm on the randomly generated low-rank matrix completion problem, the real 2D and 3D image recovery problem and the multivariate linear regression problem. Numerical results show that SVST algorithm is very competitive for low-rank matrix optimization problems in comparison with some state-of-the-art algorithms.

Suggested Citation

  • Xian Zhang & Dingtao Peng & Yanyan Su, 2024. "A singular value shrinkage thresholding algorithm for folded concave penalized low-rank matrix optimization problems," Journal of Global Optimization, Springer, vol. 88(2), pages 485-508, February.
    • Handle: RePEc:spr:jglopt:v:88:y:2024:i:2:d:10.1007_s10898-023-01322-8
    DOI: 10.1007/s10898-023-01322-8
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    References listed on IDEAS

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    1. 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.
    2. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    3. Dingtao Peng & Naihua Xiu & Jian Yu, 2017. "$$S_{1/2}$$ S 1 / 2 regularization methods and fixed point algorithms for affine rank minimization problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 543-569, July.
    4. Quan Yu & Xinzhen Zhang, 2022. "A smoothing proximal gradient algorithm for matrix rank minimization problem," Computational Optimization and Applications, Springer, vol. 81(2), pages 519-538, March.
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