Minimization of Nonsmooth Weakly Convex Function over Prox-regular Set for Robust Low-rank Matrix Recovery
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| Format: | Preprint |
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2025
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| _version_ | 1866908803821207552 |
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| author | Kume, Keita Yamada, Isao |
| author_facet | Kume, Keita Yamada, Isao |
| contents | We propose a prox-regular-type low-rank constrained nonconvex nonsmooth optimization model for Robust Low-Rank Matrix Recovery (RLRMR), i.e., estimate problem of low-rank matrix from an observed signal corrupted by outliers. For RLRMR, the $\ell_{1}$-norm has been utilized as a convex loss to detect outliers as well as to keep tractability of optimization models. Nevertheless, the $\ell_{1}$-norm is not necessarily an ideal robust loss because the $\ell_{1}$-norm tends to overpenalize entries corrupted by outliers of large magnitude. In contrast, the proposed model can employ a weakly convex function as a more robust loss, against outliers, than the $\ell_{1}$-norm. For the proposed model, we present (i) a projected variable smoothing-type algorithm applicable for the minimization of a nonsmooth weakly convex function over a prox-regular set, and (ii) a convergence analysis of the proposed algorithm in terms of stationary point. Numerical experiments demonstrate the effectiveness of the proposed model compared with the existing models that employ the $\ell_{1}$-norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17549 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimization of Nonsmooth Weakly Convex Function over Prox-regular Set for Robust Low-rank Matrix Recovery Kume, Keita Yamada, Isao Optimization and Control Signal Processing We propose a prox-regular-type low-rank constrained nonconvex nonsmooth optimization model for Robust Low-Rank Matrix Recovery (RLRMR), i.e., estimate problem of low-rank matrix from an observed signal corrupted by outliers. For RLRMR, the $\ell_{1}$-norm has been utilized as a convex loss to detect outliers as well as to keep tractability of optimization models. Nevertheless, the $\ell_{1}$-norm is not necessarily an ideal robust loss because the $\ell_{1}$-norm tends to overpenalize entries corrupted by outliers of large magnitude. In contrast, the proposed model can employ a weakly convex function as a more robust loss, against outliers, than the $\ell_{1}$-norm. For the proposed model, we present (i) a projected variable smoothing-type algorithm applicable for the minimization of a nonsmooth weakly convex function over a prox-regular set, and (ii) a convergence analysis of the proposed algorithm in terms of stationary point. Numerical experiments demonstrate the effectiveness of the proposed model compared with the existing models that employ the $\ell_{1}$-norm. |
| title | Minimization of Nonsmooth Weakly Convex Function over Prox-regular Set for Robust Low-rank Matrix Recovery |
| topic | Optimization and Control Signal Processing |
| url | https://arxiv.org/abs/2509.17549 |