🤖 AI Summary
This paper addresses the robust low-rank matrix completion problem under outliers and non-uniform sampling, formulated as a nonconvex, nonsmooth DC (Difference-of-Convex) composite optimization problem. To tackle its structural complexity, we propose the inexact Linearized Proximal Algorithm (iLPA), which—uniquely for this class of problems—establishes a verifiable Kurdyka–Łojasiewicz (KL) exponent of 1/2 and rigorously proves local R-linear convergence. Our method integrates strongly convex majorization within the DC optimization framework, balancing theoretical soundness with computational tractability. Extensive experiments on large-scale real-world datasets demonstrate that iLPA achieves significantly faster runtime than state-of-the-art methods, while attaining competitive relative error and normalized mean absolute error (NMAE) comparable to the advanced PAM algorithm—thus delivering both high efficiency and strong robustness.
📝 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 in each step an inexact minimizer of a strongly convex majorization constructed with a partial linearization of their objective functions at the current iterate, and establish the convergence of the generated iterate sequence under the Kurdyka-L""ojasiewicz (KL) property of a potential function. In particular, by leveraging the composite structure, we provide a verifiable condition for the potential function to have the KL property of exponent $1/2$ at the limit point, so for the iterate sequence to have a local R-linear convergence rate. Finally, we apply the proposed iLPA to a robust factorization model for matrix completions with outliers and non-uniform sampling, and numerical comparison with a proximal alternating minimization (PAM) method confirms iLPA yields the comparable relative errors or NMAEs within less running time, especially for large-scale real data.
