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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| Format: | Journal Article |
| Language: | English |
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01.12.2020
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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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| References | Moreau (CR22) 1965; 93 Muoi (CR23) 2015; 23 Mairal, Bach, Ponce, Sapiro (CR21) 2010; 11 Wang, Lu, Mao, Cheng (CR30) 2013; 29 Alexandrov, Becker, Deininger, Ernst, Wehder, Grasmair, von Eggeling, Thiele, Maass (CR1) 2010; 9 CR16 Muoi, Hào, Sahoo, Tang, Cong, Dang (CR26) 2018; 34 Attouch, Bolte, Svaiter (CR3) 2013; 137 Drakakis, Rickard, Fréin, Cichocki (CR9) 2008; 3 Hào, Quyen (CR14) 2011; 27 Bolte, Sabach, Teboulle (CR5) 2014; 146 Daubechies, Defrise, De Mol (CR7) 2016; 32 Engl, Hanke, Neubauer (CR10) 1996 Ito, Jin, Takeuchi (CR17) 2011; 18 Guo, Lin, Shi (CR13) 2019; 46 Lee, Seung (CR18) 1999; 401 Xu, Yin (CR31) 2013; 6 Berry, Browne, Langville, Pauca, Plemmons (CR4) 2007; 52 Hào, Quyen (CR15) 2012; 388 CR29 Lorenz, Maass, Muoi (CR20) 2012; 39 CR27 Sivananthan (CR28) 2016; 36 Engl, Kunisch, Neubauer (CR11) 1989; 5 Alexandrov, Kobarg (CR2) 2011; 27 Greene, Cagney, Krogan, Cunningham (CR12) 2008; 24 Muoi, Hào, Maass, Pidcock (CR24) 2013; 21 Brunet, Tamayo, Golub, Mesirov (CR6) 2004; 101 Muoi, Hào, Maass, Pidcock (CR25) 2016; 298 Daubechies, Defrise, Demol (CR8) 2004; 57 Lorenz (CR19) 2008; 16 K Ito (864_CR17) 2011; 18 PQ Muoi (864_CR23) 2015; 23 Y Xu (864_CR31) 2013; 6 J Mairal (864_CR21) 2010; 11 DA Lorenz (864_CR20) 2012; 39 H Attouch (864_CR3) 2013; 137 Z Guo (864_CR13) 2019; 46 J Bolte (864_CR5) 2014; 146 K Drakakis (864_CR9) 2008; 3 864_CR16 T Alexandrov (864_CR2) 2011; 27 DA Lorenz (864_CR19) 2008; 16 I Daubechies (864_CR8) 2004; 57 T Alexandrov (864_CR1) 2010; 9 J Brunet (864_CR6) 2004; 101 MW Berry (864_CR4) 2007; 52 DN Hào (864_CR15) 2012; 388 J Moreau (864_CR22) 1965; 93 W Wang (864_CR30) 2013; 29 DD Lee (864_CR18) 1999; 401 DN Hào (864_CR14) 2011; 27 I Daubechies (864_CR7) 2016; 32 D Greene (864_CR12) 2008; 24 S Sivananthan (864_CR28) 2016; 36 PQ Muoi (864_CR24) 2013; 21 864_CR27 HW Engl (864_CR10) 1996 864_CR29 PQ Muoi (864_CR26) 2018; 34 HW Engl (864_CR11) 1989; 5 PQ Muoi (864_CR25) 2016; 298 |
| References_xml | – volume: 52 start-page: 155 issue: 1 year: 2007 end-page: 173 ident: CR4 article-title: Algorithms and applications for approximate nonnegative matrix factorization publication-title: Comput. Stat. Data Analys. doi: 10.1016/j.csda.2006.11.006 – volume: 388 start-page: 593 issue: 1 year: 2012 end-page: 616 ident: CR15 article-title: Convergence rates for total variation regularization of coefficient identification problems in elliptic equations ii publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2011.11.008 – volume: 11 start-page: 19 year: 2010 end-page: 60 ident: CR21 article-title: Online learning for matrix factorization and sparse coding publication-title: J. Mach. Learn. Res. – volume: 101 start-page: 4164 issue: 12 year: 2004 end-page: 4169 ident: CR6 article-title: Metagenes and molecular pattern discovery using matrix factorization publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.0308531101 – volume: 5 start-page: 523 issue: 4 year: 1989 ident: CR11 article-title: Convergence rates for Tikhonov regularization of non-linear ill-posed problems publication-title: Inverse Problems doi: 10.1088/0266-5611/5/4/007 – ident: CR16 – volume: 34 start-page: 055007 issue: 5 year: 2018 ident: CR26 article-title: Inverse problems with nonnegative and sparse solutions: Algorithms and application to the phase retrieval problem publication-title: Inverse Problems doi: 10.1088/1361-6420/aab6c9 – volume: 146 start-page: 459 issue: 1-2 year: 2014 end-page: 494 ident: CR5 article-title: Proximal alternating linearized minimization for nonconvex and nonsmooth problems publication-title: Math. Program. doi: 10.1007/s10107-013-0701-9 – ident: CR29 – volume: 36 start-page: 141 year: 2016 end-page: 165 ident: CR28 article-title: Multi-penalty regularization in learning theory publication-title: J. Complex. doi: 10.1016/j.jco.2016.05.003 – volume: 27 start-page: 075008 year: 2011 ident: CR14 article-title: Convergence rates for total variation regularization of coefficient identification problems in elliptic equations I publication-title: Inverse Problems doi: 10.1088/0266-5611/27/7/075008 – volume: 39 start-page: 437 year: 2012 end-page: 463 ident: CR20 article-title: Gradient descent methods based on quadratic approximations of Tikhonov functionals with sparsity constraints: theory and numerical comparison of stepsize rules publication-title: Electron. Trans. Numer. Anal. – ident: CR27 – volume: 298 start-page: 105 year: 2016 end-page: 122 ident: CR25 article-title: Descent gradient methods for nonsmooth minimization problems in ill-posed problems publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2015.11.039 – volume: 32 start-page: 104001 issue: 10 year: 2016 ident: CR7 article-title: Sparsity-enforcing regularisation and ISTA revisited publication-title: Inverse Problems doi: 10.1088/0266-5611/32/10/104001 – volume: 24 start-page: 1722 issue: 15 year: 2008 end-page: 1728 ident: CR12 article-title: Ensemble non-negative matrix factorization methods for clustering protein–protein interactions publication-title: Bioinformatics doi: 10.1093/bioinformatics/btn286 – volume: 9 start-page: 6535 issue: 12 year: 2010 end-page: 6546 ident: CR1 article-title: Spatial segmentation of imaging mass spectrometry data with edge-preserving image denoising and clustering publication-title: J. Proteome Res. doi: 10.1021/pr100734z – volume: 23 start-page: 1366 issue: 8 year: 2015 end-page: 1387 ident: CR23 article-title: Reconstructing conductivity coefficients based on sparsity regularization and measured data in electrical impedance tomography publication-title: Inverse Problems in Science and Engineering doi: 10.1080/17415977.2015.1018678 – volume: 57 start-page: 1413 year: 2004 end-page: 1541 ident: CR8 article-title: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.20042 – volume: 46 start-page: 478 year: 2019 end-page: 499 ident: CR13 article-title: Distributed learning with multi-penalty regularization publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2017.06.001 – volume: 93 start-page: 273 year: 1965 end-page: 299 ident: CR22 article-title: Proximité et dualité dans un espace hilbertien publication-title: Bulletin de la Société Mathématique de France doi: 10.24033/bsmf.1625 – volume: 21 start-page: 665 issue: 5 year: 2013 end-page: 693 ident: CR24 article-title: Semismooth Newton and quasi-Newton methods in weighted l1-regularization publication-title: Journal of Inverse and Ill-Posed Problems doi: 10.1515/jip-2013-0031 – volume: 29 start-page: 065018 issue: 6 year: 2013 ident: CR30 article-title: Multi-parameter Tikhonov regularization with the l0 sparsity constraint publication-title: Inverse Problems doi: 10.1088/0266-5611/29/6/065018 – volume: 18 start-page: 31 issue: 1 year: 2011 end-page: 46 ident: CR17 article-title: Multi-parameter Tikhonov regularization publication-title: Methods and Applications of Analysis doi: 10.4310/MAA.2011.v18.n1.a2 – volume: 6 start-page: 1758 issue: 3 year: 2013 end-page: 1789 ident: CR31 article-title: A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion publication-title: SIAM J. Imaging Sci. doi: 10.1137/120887795 – volume: 137 start-page: 91 issue: 1-2 year: 2013 end-page: 129 ident: CR3 article-title: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods publication-title: Math. Program. doi: 10.1007/s10107-011-0484-9 – year: 1996 ident: CR10 publication-title: Regularization of inverse problems doi: 10.1007/978-94-009-1740-8 – volume: 27 start-page: i230 issue: 13 year: 2011 end-page: i238 ident: CR2 article-title: Efficient spatial segmentation of large imaging mass spectrometry datasets with spatially aware clustering publication-title: Bioinformatics doi: 10.1093/bioinformatics/btr246 – volume: 401 start-page: 788 issue: 6755 year: 1999 end-page: 791 ident: CR18 article-title: Learning the parts of objects by non-negative matrix factorization publication-title: Nature doi: 10.1038/44565 – volume: 3 start-page: 1853 year: 2008 end-page: 1870 ident: CR9 article-title: Analysis of financial data using non-negative matrix factorization publication-title: Int. Math. Forum – volume: 16 start-page: 463 year: 2008 end-page: 478 ident: CR19 article-title: Convergence rates and source conditions for Tikhonov regularization with sparsity constraints publication-title: J. Inv Ill-posed problems doi: 10.1515/JIIP.2008.025 – volume: 32 start-page: 104001 issue: 10 year: 2016 ident: 864_CR7 publication-title: Inverse Problems doi: 10.1088/0266-5611/32/10/104001 – volume: 93 start-page: 273 year: 1965 ident: 864_CR22 publication-title: Bulletin de la Société Mathématique de France doi: 10.24033/bsmf.1625 – volume: 5 start-page: 523 issue: 4 year: 1989 ident: 864_CR11 publication-title: Inverse Problems doi: 10.1088/0266-5611/5/4/007 – volume: 21 start-page: 665 issue: 5 year: 2013 ident: 864_CR24 publication-title: Journal of Inverse and Ill-Posed Problems doi: 10.1515/jip-2013-0031 – volume: 36 start-page: 141 year: 2016 ident: 864_CR28 publication-title: J. Complex. doi: 10.1016/j.jco.2016.05.003 – volume-title: Regularization of inverse problems year: 1996 ident: 864_CR10 doi: 10.1007/978-94-009-1740-8 – volume: 39 start-page: 437 year: 2012 ident: 864_CR20 publication-title: Electron. Trans. Numer. Anal. – volume: 6 start-page: 1758 issue: 3 year: 2013 ident: 864_CR31 publication-title: SIAM J. Imaging Sci. doi: 10.1137/120887795 – ident: 864_CR16 – volume: 9 start-page: 6535 issue: 12 year: 2010 ident: 864_CR1 publication-title: J. Proteome Res. doi: 10.1021/pr100734z – volume: 101 start-page: 4164 issue: 12 year: 2004 ident: 864_CR6 publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.0308531101 – volume: 23 start-page: 1366 issue: 8 year: 2015 ident: 864_CR23 publication-title: Inverse Problems in Science and Engineering doi: 10.1080/17415977.2015.1018678 – volume: 34 start-page: 055007 issue: 5 year: 2018 ident: 864_CR26 publication-title: Inverse Problems doi: 10.1088/1361-6420/aab6c9 – volume: 29 start-page: 065018 issue: 6 year: 2013 ident: 864_CR30 publication-title: Inverse Problems doi: 10.1088/0266-5611/29/6/065018 – volume: 18 start-page: 31 issue: 1 year: 2011 ident: 864_CR17 publication-title: Methods and Applications of Analysis doi: 10.4310/MAA.2011.v18.n1.a2 – ident: 864_CR29 – volume: 52 start-page: 155 issue: 1 year: 2007 ident: 864_CR4 publication-title: Comput. Stat. Data Analys. doi: 10.1016/j.csda.2006.11.006 – volume: 24 start-page: 1722 issue: 15 year: 2008 ident: 864_CR12 publication-title: Bioinformatics doi: 10.1093/bioinformatics/btn286 – volume: 11 start-page: 19 year: 2010 ident: 864_CR21 publication-title: J. Mach. Learn. Res. – volume: 57 start-page: 1413 year: 2004 ident: 864_CR8 publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.20042 – volume: 146 start-page: 459 issue: 1-2 year: 2014 ident: 864_CR5 publication-title: Math. Program. doi: 10.1007/s10107-013-0701-9 – volume: 46 start-page: 478 year: 2019 ident: 864_CR13 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2017.06.001 – volume: 401 start-page: 788 issue: 6755 year: 1999 ident: 864_CR18 publication-title: Nature doi: 10.1038/44565 – volume: 137 start-page: 91 issue: 1-2 year: 2013 ident: 864_CR3 publication-title: Math. Program. doi: 10.1007/s10107-011-0484-9 – volume: 27 start-page: 075008 year: 2011 ident: 864_CR14 publication-title: Inverse Problems doi: 10.1088/0266-5611/27/7/075008 – volume: 388 start-page: 593 issue: 1 year: 2012 ident: 864_CR15 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2011.11.008 – volume: 298 start-page: 105 year: 2016 ident: 864_CR25 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2015.11.039 – volume: 27 start-page: i230 issue: 13 year: 2011 ident: 864_CR2 publication-title: Bioinformatics doi: 10.1093/bioinformatics/btr246 – volume: 3 start-page: 1853 year: 2008 ident: 864_CR9 publication-title: Int. Math. Forum – ident: 864_CR27 – volume: 16 start-page: 463 year: 2008 ident: 864_CR19 publication-title: J. Inv Ill-posed problems doi: 10.1515/JIIP.2008.025 |
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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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