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

• URL: http://arxiv.org/abs/2303.16822v4
• Date: Fri, 7 Jun 2024 10:12:12 GMT
• Title: An inexact LPA for DC composite optimization and application to matrix completions with outliers
• Authors: Ting Tao, Ruyu Liu, Shaohua Pan,
• Abstract summary: This paper concerns a class of composite optimization problems. By leveraging the composite structure, we provide a condition for the potential function to have the KL property of $1/2$ at the iterate sequence.
• Score: 5.746154410100363
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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