An inexact LPA for DC composite optimization and application to matrix completions with outliers
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| Format: | Preprint |
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2023
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| _version_ | 1866909827527081984 |
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| author | Tao, Ting Liu, Ruyu Pan, Shaohua |
| author_facet | Tao, Ting Liu, Ruyu Pan, Shaohua |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_16822 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An inexact LPA for DC composite optimization and application to matrix completions with outliers Tao, Ting Liu, Ruyu Pan, Shaohua Optimization and Control Machine Learning 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. |
| title | An inexact LPA for DC composite optimization and application to matrix completions with outliers |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2303.16822 |