{"ID":2867596,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.17549","arxiv_id":"2509.17549","title":"Minimization of Nonsmooth Weakly Convex Function over Prox-regular Set for Robust Low-rank Matrix Recovery","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2509.17549","url_pdf":"https://arxiv.org/pdf/2509.17549v2","authors":"[\"Keita Kume\",\"Isao Yamada\"]","published":"2025-09-22T09:08:35Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"eess.SP\"]","methods":"[]","has_code":false}