Minimization of Nonsmooth Weakly Convex Function over Prox-regular Set for Robust Low-rank Matrix Recovery

2025/09/22 by Kume, Keita, Yamada, Isao
#FOS: Electrical engineering #FOS: Mathematics #Optimization and Control (math.OC) #Signal Processing (eess.SP) #electronic engineering #information engineering

paper · doi:10.48550/arxiv.2509.17549

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 ℓ₁-norm has been utilized as a convex loss to detect outliers as well as to keep tractability of optimization models. Nevertheless, the ℓ₁-norm is not necessarily an ideal robust loss because the ℓ₁-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 ℓ₁-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 ℓ₁-norm.

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