Convergence of proximal algorithms with stepsize controls for non-linear inverse problems and application to sparse non-negative matrix factorization
We consider a general ill-posed inverse problem in a Hilbert space setting by minimizing a misfit functional coupling with a multi-penalty regularization for stabilization. For solving this minimization problem, we investigate two proximal algorithms with stepsize controls: a proximal fixed point al...
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| Published in: | Numerical algorithms Vol. 85; no. 4; pp. 1255 - 1279 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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01.12.2020
Springer Nature B.V |
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| ISSN: | 1017-1398, 1572-9265 |
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| Abstract | We consider a general ill-posed inverse problem in a Hilbert space setting by minimizing a misfit functional coupling with a multi-penalty regularization for stabilization. For solving this minimization problem, we investigate two proximal algorithms with stepsize controls: a proximal fixed point algorithm and an alternating proximal algorithm. We prove the decrease of the objective functional and the convergence of both update schemes to a stationary point under mild conditions on the stepsizes. These algorithms are then applied to the sparse and non-negative matrix factorization problems. Based on a priori information of non-negativity and sparsity of the exact solution, the problem is regularized by corresponding terms. In both cases, the implementation of our proposed algorithms is straight-forward since the evaluation of the proximal operators in these problems can be done explicitly. Finally, we test the proposed algorithms for the non-negative sparse matrix factorization problem with both simulated and real-world data and discuss reconstruction performance, convergence, as well as achieved sparsity. |
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| AbstractList | We consider a general ill-posed inverse problem in a Hilbert space setting by minimizing a misfit functional coupling with a multi-penalty regularization for stabilization. For solving this minimization problem, we investigate two proximal algorithms with stepsize controls: a proximal fixed point algorithm and an alternating proximal algorithm. We prove the decrease of the objective functional and the convergence of both update schemes to a stationary point under mild conditions on the stepsizes. These algorithms are then applied to the sparse and non-negative matrix factorization problems. Based on a priori information of non-negativity and sparsity of the exact solution, the problem is regularized by corresponding terms. In both cases, the implementation of our proposed algorithms is straight-forward since the evaluation of the proximal operators in these problems can be done explicitly. Finally, we test the proposed algorithms for the non-negative sparse matrix factorization problem with both simulated and real-world data and discuss reconstruction performance, convergence, as well as achieved sparsity. |
| Author | Pham, Quy Muoi Hào, Dinh Nho Lachmund, Delf |
| Author_xml | – sequence: 1 givenname: Quy Muoi orcidid: 0000-0001-8031-9078 surname: Pham fullname: Pham, Quy Muoi email: phamquymuoi@gmail.com organization: Department of Mathematics, The University of Danang - University of Science and Education – sequence: 2 givenname: Delf surname: Lachmund fullname: Lachmund, Delf organization: Center for Industrial Mathematics, University of Bremen – sequence: 3 givenname: Dinh Nho surname: Hào fullname: Hào, Dinh Nho organization: Hanoi Institute of Mathematics, VAST |
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| Cites_doi | 10.1016/j.csda.2006.11.006 10.1016/j.jmaa.2011.11.008 10.1073/pnas.0308531101 10.1088/0266-5611/5/4/007 10.1088/1361-6420/aab6c9 10.1007/s10107-013-0701-9 10.1016/j.jco.2016.05.003 10.1088/0266-5611/27/7/075008 10.1016/j.cam.2015.11.039 10.1088/0266-5611/32/10/104001 10.1093/bioinformatics/btn286 10.1021/pr100734z 10.1080/17415977.2015.1018678 10.1002/cpa.20042 10.1016/j.acha.2017.06.001 10.24033/bsmf.1625 10.1515/jip-2013-0031 10.1088/0266-5611/29/6/065018 10.4310/MAA.2011.v18.n1.a2 10.1137/120887795 10.1007/s10107-011-0484-9 10.1007/978-94-009-1740-8 10.1093/bioinformatics/btr246 10.1038/44565 10.1515/JIIP.2008.025 |
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| Keywords | Non-smooth Optimization Proximal fixed point iteration Alternating proximal iteration Sparse non-negative matrix factorization Non-linear inverse problems Sparsity regularization Sparse matrix factorization 65Kxx Non-negative sparsity regularization 49Mxx |
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| SubjectTerms | Algebra Algorithms Computer Science Convergence Exact solutions Factorization Hilbert space Inverse problems Mathematical analysis Nonlinear control Numeric Computing Numerical Analysis Operators (mathematics) Original Paper Regularization Regularization methods Sparse matrices Sparsity Theory of Computation |
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| Title | Convergence of proximal algorithms with stepsize controls for non-linear inverse problems and application to sparse non-negative matrix factorization |
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| Volume | 85 |
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