An inexact LPA for DC composite optimization and application to matrix completions with outliers

TL;DR

The paper tackles DC composite optimization of the form $\min_x \Phi(x)=\vartheta_1(F(x))-\vartheta_2(G(x))+h(x)$, a challenging nonconvex and nonsmooth class with applications to robust low-rank matrix recovery. It introduces an inexact linearized proximal algorithm (iLPA) that solves inexactly majorized proximal subproblems built from partial linearizations, and proves global convergence via a KL-based potential $\Xi$, with a verifiable condition yielding local $R$-linear convergence when the KL exponent is $1/2$. A dual proximal point solver with semismooth Newton (dPPASN) efficiently handles subproblems, enabling scalable performance. The approach is validated numerically on DC problems with nonsmooth components and on matrix completion with outliers under nonuniform sampling, showing competitive or superior accuracy and substantially faster runtimes than alternative methods such as nmBDCA and PAM. Overall, the framework provides a practical, theoretically sound path to globally convergent solutions for complex DC composite problems with concrete benefits in robust matrix recovery tasks.

Abstract

This paper concerns a class of DC composite optimization problems which, as an extension of convex composite optimization problems and DC programs with nonsmooth components, often arises in robust factorization models of low-rank matrix recovery. For this class of nonconvex and nonsmooth problems, we propose an inexact linearized proximal algorithm (iLPA) by computing at each step an inexact minimizer of a strongly convex majorization constructed with a partial linearization of their objective functions at the current iterate. We establish the full convergence of the generated iterate sequence under the Kurdyka-Łöjasiewicz (KL) property of a potential function, and employ the composite structure to provide a verifiable condition for the potential function to satisfy the KL property of exponent $1/2$ at the limit point, so for the iterate sequence to have a local R-linear convergence rate. This condition is weaker than the one provided in \cite[Theorem 3.2]{LiPong18} for identifying the KL property of exponent $p\in[0,1)$ for a general composite function. The proposed iLPA is applied to a robust factorization model for matrix completion with outliers and non-uniform sampling, and numerical comparisons with the Polyak subgradient method and a proximal alternating minimization (PAM) method validate its efficiency.

Figures (7)

• Figure 1: The performance of iLPA with $\mu_k=c_{\mu}k^{-1}$ for different $c_{\mu}$ and that of its inner solver dPPASN
• Figure 2: The number of subproblems for each step of iLPA with $\mu_k=10^6 k^{-1}$ and the maximum number of iterations of dPPASN
• Figure 3: The relative error and sparsity ratio curves of iLPA under noise of type IV with $n_1=n_2=1000,r^*=10,SR=0.15$
• Figure 4: Iteration and time curves of iLPA, PAM and subGM under noise of type IV with $n_1=n_2=3000, r^*=10$ and $SR=0.15$
• Figure 5: The relative error and rank curves of iLPA, PAM and subGM as the parameter $c_{\lambda}$ or $\lambda$ increases

Theorems & Definitions (33)

• Example 1.1
• Example 1.2
• Example 1.3
• Definition 2.1
• Remark 2.1
• Definition 2.2
• Lemma 2.1
• Definition 2.3
• Definition 2.4
• Remark 2.2