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

Keita Kume, Isao Yamada

TL;DR

The authors address robust low-rank matrix recovery from outlier-corrupted measurements by formulating a nonconvex, nonsmooth optimization that minimizes a weakly convex loss over a prox-regular low-rank set $\mathfrak{L}_{r,\sigma}$. They replace the traditional $\ell_1$-loss with a weakly convex loss $\ell$ and solve $\min_{X\in \mathfrak{L}_{r,\sigma}} \sum_{i=1}^m \ell([\mathbf{y}]_i-\mathcal{A}_i(X))$, using a projected variable smoothing algorithm based on the Moreau envelope to obtain a Lipschitz gradient and a stationarity measure $\mathcal{M}_{\gamma}^{F,\iota_C}$. The method provides asymptotic convergence to stationary points and, in synthetic experiments, SCAD-based weakly convex losses substantially improve recovery accuracy over $\ell_1$-based and other benchmarks, particularly under severe outliers. The work offers a principled, convergent approach for robust low-rank recovery with potential impact across phase retrieval, RPCA, and matrix completion contexts.

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.

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

TL;DR

The authors address robust low-rank matrix recovery from outlier-corrupted measurements by formulating a nonconvex, nonsmooth optimization that minimizes a weakly convex loss over a prox-regular low-rank set . They replace the traditional -loss with a weakly convex loss and solve , using a projected variable smoothing algorithm based on the Moreau envelope to obtain a Lipschitz gradient and a stationarity measure . The method provides asymptotic convergence to stationary points and, in synthetic experiments, SCAD-based weakly convex losses substantially improve recovery accuracy over -based and other benchmarks, particularly under severe outliers. The work offers a principled, convergent approach for robust low-rank recovery with potential impact across phase retrieval, RPCA, and matrix completion contexts.

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

Paper Structure

This paper contains 5 sections, 3 theorems, 15 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary on Nonsmooth Analysis
  3. Projected variable smoothing algorithm
  4. Numerical Experiments on Synthetic Data
  5. Concluding Remarks

Key Result

Lemma 3.1

$\bm{x}^{\star} \in \mathcal{X}$ is a stationary point of Problem problem:origin, i.e., $\partial_{F}(F+\iota_{C})(\bm{x}^{\star}) \ni \bm{0}$, if and only if $\mathcal{M}_{\gamma}^{F,\iota_{C}}(\bm{x}^{\star}) = 0$ holds for some $\gamma \in \mathbb{R}_{++}$.

Theorems & Definitions (6)

  • Remark 1.3: Reformulation of the model \ref{['eq:proposed_model']} into Problem \ref{['problem:origin']}
  • Definition 2.1: Subdifferential Rockafellar-Wets98
  • Lemma 3.1: Stationarity characterization via $\mathcal{M}_{\gamma}^{F,\iota_{C}}$
  • Theorem 3.3: Asymptotic property of and $\mathcal{M}_{\gamma}^{{}^{\mu}g\circ\mathfrak{S},\phi}$
  • Theorem 3.4: Convergence analysis of Alg. \ref{['alg:proposed']}
  • Remark 3.5: Extension of Alg. \ref{['alg:proposed']} to the minimization of $F+\phi$ with a prox-regular function $\phi$