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Low-Rank Matrix Recovery Via Nonconvex Optimization Methods with Application to Errors-in-Variables Matrix Regression

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  • Xin Li

    (Northwest University)

  • Dongya Wu

    (Northwest University)

Abstract

We consider the nonconvex regularized method for low-rank matrix recovery problems. Under suitable regularity conditions on the nonconvex loss function and the regularizer, we provide the recovery bound for any stationary point of the nonconvex method via separating singular values of the parameter matrix into larger and smaller ones. In this way, the established recovery bound can be much tighter than that of the convex nuclear norm regularized method when some of the singular values are larger than a threshold defined by the nonconvex regularizer. In addition, we consider the errors-in-variables matrix regression as an application of the nonconvex method. Probabilistic consequences and the advantage of the nonconvex method are demonstrated through verifying the regularity conditions for specific models with additive noise and missing data.

Suggested Citation

  • Xin Li & Dongya Wu, 2025. "Low-Rank Matrix Recovery Via Nonconvex Optimization Methods with Application to Errors-in-Variables Matrix Regression," Journal of Optimization Theory and Applications, Springer, vol. 205(3), pages 1-27, June.
    • Handle: RePEc:spr:joptap:v:205:y:2025:i:3:d:10.1007_s10957-025-02660-1
    DOI: 10.1007/s10957-025-02660-1
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    References listed on IDEAS

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    1. Wu, Jie & Zheng, Zemin & Li, Yang & Zhang, Yi, 2020. "Scalable interpretable learning for multi-response error-in-variables regression," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    2. Radu-Alexandru Dragomir & Alexandre d’Aspremont & Jérôme Bolte, 2021. "Quartic First-Order Methods for Low-Rank Minimization," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 341-363, May.
    3. Xin Li & Dongya Wu, 2024. "Low-rank matrix estimation via nonconvex optimization methods in multi-response errors-in-variables regression," Journal of Global Optimization, Springer, vol. 88(1), pages 79-114, January.
    4. Shuvomoy Das Gupta & Bartolomeo Stellato & Bart P. G. Parys, 2024. "Exterior-Point Optimization for Sparse and Low-Rank Optimization," Journal of Optimization Theory and Applications, Springer, vol. 202(2), pages 795-833, August.
    5. Hua Zhou & Lexin Li & Hongtu Zhu, 2013. "Tensor Regression with Applications in Neuroimaging Data Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(502), pages 540-552, June.
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