A unified framework for nonconvex nonsmooth sparse and low-rank decomposition by majorization-minimization algorithm
Recovering a low-rank matrix and a sparse matrix from an observed matrix, known as sparse and low-rank decomposition (SLRD), is becoming a hot topic in recent years. The most popular model for SLRD is to use the ℓ1 norm and nuclear norm for the sparse and low-rank approximation. Since this convex mo...
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| Veröffentlicht in: | Journal of the Franklin Institute Jg. 359; H. 16; S. 9376 - 9400 |
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01.11.2022
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| ISSN: | 0016-0032, 1879-2693 |
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| Abstract | Recovering a low-rank matrix and a sparse matrix from an observed matrix, known as sparse and low-rank decomposition (SLRD), is becoming a hot topic in recent years. The most popular model for SLRD is to use the ℓ1 norm and nuclear norm for the sparse and low-rank approximation. Since this convex model has certain limitations, various nonconvex models have been explored and found to be very promising. In this paper, we introduce a generalized nonconvex nonsmooth model for SLRD which covers a wide range of nonconvex surrogate functions that are continuous, concave and monotonically increasing on [0,∞) to approximate both the ℓ0 norm and the rank function, such as ℓp norm (0<p<1), Logarithm, Geman, SCAD and MCP functions. The choice of the nonconvex surrogates for the sparse and low-rank components can be different. Due to the nonconvexity and extensive options of the surrogates, the optimization problem is untractable. Based on the majorization-minimization (MM) algorithm, we propose a unified framework named MM-ADMM algorithm to solve this problem, which can be applied to all eligible surrogates as long as their supergradients are available. The constrained majorizing problems established under the MM framework can be easily solved by the alternating direction method of multipliers (ADMM). The theoretical convergence properties are investigated and proved, including the convergence of the sequence of objective function values generated by the designed algorithm and a weak convergence result related to the inner ADMM-iterations. Experiments on the synthetic data and real-world applications demonstrate the effectiveness of our designed MM-ADMM algorithm. |
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| AbstractList | Recovering a low-rank matrix and a sparse matrix from an observed matrix, known as sparse and low-rank decomposition (SLRD), is becoming a hot topic in recent years. The most popular model for SLRD is to use the ℓ1 norm and nuclear norm for the sparse and low-rank approximation. Since this convex model has certain limitations, various nonconvex models have been explored and found to be very promising. In this paper, we introduce a generalized nonconvex nonsmooth model for SLRD which covers a wide range of nonconvex surrogate functions that are continuous, concave and monotonically increasing on [0,∞) to approximate both the ℓ0 norm and the rank function, such as ℓp norm (0<p<1), Logarithm, Geman, SCAD and MCP functions. The choice of the nonconvex surrogates for the sparse and low-rank components can be different. Due to the nonconvexity and extensive options of the surrogates, the optimization problem is untractable. Based on the majorization-minimization (MM) algorithm, we propose a unified framework named MM-ADMM algorithm to solve this problem, which can be applied to all eligible surrogates as long as their supergradients are available. The constrained majorizing problems established under the MM framework can be easily solved by the alternating direction method of multipliers (ADMM). The theoretical convergence properties are investigated and proved, including the convergence of the sequence of objective function values generated by the designed algorithm and a weak convergence result related to the inner ADMM-iterations. Experiments on the synthetic data and real-world applications demonstrate the effectiveness of our designed MM-ADMM algorithm. |
| Author | Zheng, Qian-Zhen Xu, Ping-Feng |
| Author_xml | – sequence: 1 givenname: Qian-Zhen surname: Zheng fullname: Zheng, Qian-Zhen organization: School of Mathematics and Statistics, Changchun University of Technology, Changchun 130012, China – sequence: 2 givenname: Ping-Feng orcidid: 0000-0002-4721-9996 surname: Xu fullname: Xu, Ping-Feng email: xupf900@nenu.edu.cn organization: Academy for Advanced Interdisciplinary Studies, Northeast Normal University, Changchun 130024, China |
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| Cites_doi | 10.1007/s12204-016-1765-5 10.1155/2014/656074 10.1109/TPAMI.2011.282 10.1109/TPAMI.2015.2465956 10.1016/j.jfranklin.2019.09.017 10.1109/JSEN.2020.2974725 10.1109/TSP.2012.2208955 10.1109/TIP.2016.2599290 10.1109/ACCESS.2018.2880454 10.1016/j.jfranklin.2018.12.013 10.1023/A:1004603514434 10.1023/A:1017522623963 10.1145/2674559 10.1007/s11263-016-0930-5 10.1109/TIP.2015.2511584 10.1198/0003130042836 10.1198/016214501753382273 10.1016/j.cviu.2013.11.009 10.1016/j.neucom.2017.12.034 10.1007/s10994-014-5469-5 10.1007/s11042-022-12509-8 10.1137/15M1027528 10.1109/TSP.2020.3011024 10.1109/TPAMI.2019.2929043 10.1109/TCSVT.2019.2908833 10.1145/1970392.1970395 10.1561/2200000016 10.1016/j.neunet.2016.09.005 10.1016/j.jfranklin.2020.03.032 10.1109/ACCESS.2018.2872688 10.1109/TIP.2015.2419084 10.1093/biomet/asaa066 10.1007/s13042-018-0814-9 10.1016/j.patcog.2015.01.024 10.1093/biomet/ast036 10.1137/090761793 10.1109/TSP.2019.2940121 10.1007/s11425-015-5081-6 |
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