A generalized nonconvex algorithm framework for low-rank and sparse matrix decomposition
The low-rank and sparse matrix decomposition problem is a hot and challenging problem in computer science. In this paper, we consider it as a nonconvex relaxation optimization problem by using a family of nonconvex functions to approximate the rank function and the -norm in low-rank and sparse matri...
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| Vydáno v: | Applied intelligence (Dordrecht, Netherlands) Ročník 55; číslo 16; s. 1085 |
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| Jazyk: | angličtina |
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01.11.2025
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| Abstract | The low-rank and sparse matrix decomposition problem is a hot and challenging problem in computer science. In this paper, we consider it as a nonconvex relaxation optimization problem by using a family of nonconvex functions to approximate the rank function and the
-norm in low-rank and sparse matrix decomposition problem, namely, generalized low-rank and sparse matrix decomposition problem. The essence of this paper is to develop an adaptive algorithm framework with parameters updating for the nonconvex relaxation problem. Firstly, we prove the equivalence between the generalized low-rank and sparse matrix decomposition problem and the regularization generalized low-rank and sparse matrix decomposition problem. This means that the optimal solution of generalized low-rank and sparse matrix decomposition problem can be exactly obtained by solving its regularization minimization problem. Secondly, we present a tractable nonconvex algorithm framework to solve the regularization generalized low-rank and sparse matrix decomposition problem. The convergence analysis of the algorithm framework is provided. More importantly, we also define a very powerful parameter-setting strategy to adapt the optimal parameters in iteration of the proposed algorithm framework. Finally, we test the proposed algorithms on some random low-rank and sparse matrix decomposition problems, and the numerical results verified the effectiveness of the proposed algorithms. In addition, we also extend the proposed algorithms to the image denoising and background modeling from surveillance video. |
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| AbstractList | The low-rank and sparse matrix decomposition problem is a hot and challenging problem in computer science. In this paper, we consider it as a nonconvex relaxation optimization problem by using a family of nonconvex functions to approximate the rank function and the -norm in low-rank and sparse matrix decomposition problem, namely, generalized low-rank and sparse matrix decomposition problem. The essence of this paper is to develop an adaptive algorithm framework with parameters updating for the nonconvex relaxation problem. Firstly, we prove the equivalence between the generalized low-rank and sparse matrix decomposition problem and the regularization generalized low-rank and sparse matrix decomposition problem. This means that the optimal solution of generalized low-rank and sparse matrix decomposition problem can be exactly obtained by solving its regularization minimization problem. Secondly, we present a tractable nonconvex algorithm framework to solve the regularization generalized low-rank and sparse matrix decomposition problem. The convergence analysis of the algorithm framework is provided. More importantly, we also define a very powerful parameter-setting strategy to adapt the optimal parameters in iteration of the proposed algorithm framework. Finally, we test the proposed algorithms on some random low-rank and sparse matrix decomposition problems, and the numerical results verified the effectiveness of the proposed algorithms. In addition, we also extend the proposed algorithms to the image denoising and background modeling from surveillance video. The low-rank and sparse matrix decomposition problem is a hot and challenging problem in computer science. In this paper, we consider it as a nonconvex relaxation optimization problem by using a family of nonconvex functions to approximate the rank function and the -norm in low-rank and sparse matrix decomposition problem, namely, generalized low-rank and sparse matrix decomposition problem. The essence of this paper is to develop an adaptive algorithm framework with parameters updating for the nonconvex relaxation problem. Firstly, we prove the equivalence between the generalized low-rank and sparse matrix decomposition problem and the regularization generalized low-rank and sparse matrix decomposition problem. This means that the optimal solution of generalized low-rank and sparse matrix decomposition problem can be exactly obtained by solving its regularization minimization problem. Secondly, we present a tractable nonconvex algorithm framework to solve the regularization generalized low-rank and sparse matrix decomposition problem. The convergence analysis of the algorithm framework is provided. More importantly, we also define a very powerful parameter-setting strategy to adapt the optimal parameters in iteration of the proposed algorithm framework. Finally, we test the proposed algorithms on some random low-rank and sparse matrix decomposition problems, and the numerical results verified the effectiveness of the proposed algorithms. In addition, we also extend the proposed algorithms to the image denoising and background modeling from surveillance video. |
| ArticleNumber | 1085 |
| Author | Zhang, Lijun Xue, Shengli Cui, Angang He, Haizhen |
| Author_xml | – sequence: 1 givenname: Angang orcidid: 0000-0001-6528-1446 surname: Cui fullname: Cui, Angang email: cuiangang@yulinu.edu.cn organization: School of Mathematics and Statistics, Yulin University – sequence: 2 givenname: Lijun surname: Zhang fullname: Zhang, Lijun organization: School of Marine Science and Technology, Northwestern Polytechnical University – sequence: 3 givenname: Haizhen surname: He fullname: He, Haizhen organization: School of Foreign Languages, Yulin University – sequence: 4 givenname: Shengli surname: Xue fullname: Xue, Shengli organization: School of Mathematics and Statistics, Yulin University |
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| Cites_doi | 10.1137/080738970 10.1007/s11760-018-1367-9 10.1109/TSP.2014.2298839 10.1002/cpa.20042 10.1109/TCSVT.2019.2908833 10.1002/nla.2055 10.1109/TNNLS.2012.2197412 10.1016/j.jvcir.2012.10.006 10.1109/CVPR.2011.5995484 10.4310/CMS.2017.v15.n2.a9 10.1109/TIT.2015.2429611 10.1016/j.ins.2018.08.037 10.1145/1970392.1970395 10.1145/1871437.1871475 10.1007/s10589-017-9898-5 10.1109/TNNLS.2019.2921404 10.1016/j.neucom.2015.10.037 |
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| SubjectTerms | Adaptive algorithms Algorithms Approximation Artificial Intelligence Computer Science Decomposition Machines Manufacturing Mechanical Engineering Optimization Parameters Processes Regularization Sparse matrices Sparsity |
| Title | A generalized nonconvex algorithm framework for low-rank and sparse matrix decomposition |
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