Solution sets of three sparse optimization problems for multivariate regression
In multivariate regression analysis, a coefficient matrix is used to relate multiple response variables to regressor variables in a noisy linear system from given data. Optimization is a natural approach to find such coefficient matrix with few nonzero rows. However, the relationship of most group s...
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| Published in | Journal of global optimization Vol. 87; no. 2-4; pp. 347 - 371 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.11.2023
Springer |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0925-5001 1573-2916 |
| DOI | 10.1007/s10898-021-01124-w |
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| Abstract | In multivariate regression analysis, a coefficient matrix is used to relate multiple response variables to regressor variables in a noisy linear system from given data. Optimization is a natural approach to find such coefficient matrix with few nonzero rows. However, the relationship of most group sparse optimization models with cardinality penalty or cardinality constraints is not clear. In this paper, we give a comprehensive description of the relationship between three widely used group sparse optimization problems with cardinality terms: (i) the number of nonzero rows is minimized subject to an error tolerance for regression; (ii) the error for regression is minimized subject to a row cardinality constraint; (iii) the sum of the number of nonzero rows and error for regression is minimized. The first two problems have convex constraints and cardinality constraints respectively, while the third one is an unconstrained optimization problem with a cardinality penalty. We provide sufficient conditions under which the three optimization problems have the same global minimizers. Moreover, we analyze the relationship of stationary points and local minimizers of the three problems. Finally, we use two examples to illustrate our theoretical results for finding solutions of constrained optimization problems involving cardinality terms by unconstrained optimization problems with penalty functions. |
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| AbstractList | In multivariate regression analysis, a coefficient matrix is used to relate multiple response variables to regressor variables in a noisy linear system from given data. Optimization is a natural approach to find such coefficient matrix with few nonzero rows. However, the relationship of most group sparse optimization models with cardinality penalty or cardinality constraints is not clear. In this paper, we give a comprehensive description of the relationship between three widely used group sparse optimization problems with cardinality terms: (i) the number of nonzero rows is minimized subject to an error tolerance for regression; (ii) the error for regression is minimized subject to a row cardinality constraint; (iii) the sum of the number of nonzero rows and error for regression is minimized. The first two problems have convex constraints and cardinality constraints respectively, while the third one is an unconstrained optimization problem with a cardinality penalty. We provide sufficient conditions under which the three optimization problems have the same global minimizers. Moreover, we analyze the relationship of stationary points and local minimizers of the three problems. Finally, we use two examples to illustrate our theoretical results for finding solutions of constrained optimization problems involving cardinality terms by unconstrained optimization problems with penalty functions. |
| Audience | Academic |
| Author | Chen, Xiaojun Pan, Lili Xiu, Naihua |
| Author_xml | – sequence: 1 givenname: Xiaojun surname: Chen fullname: Chen, Xiaojun email: maxjchen@polyu.edu.hk organization: Department of Applied Mathematics, The Hong Kong Polytechnic University – sequence: 2 givenname: Lili surname: Pan fullname: Pan, Lili organization: Department of Mathematics, Shandong University of Technology – sequence: 3 givenname: Naihua surname: Xiu fullname: Xiu, Naihua organization: Department of Applied Mathematics, Beijing Jiaotong University |
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| CitedBy_id | crossref_primary_10_1007_s10957_024_02381_x crossref_primary_10_12677_ORF_2023_136734 crossref_primary_10_12677_ORF_2023_136745 crossref_primary_10_1287_moor_2023_0295 crossref_primary_10_1007_s10589_023_00465_4 crossref_primary_10_1016_j_neucom_2024_128267 crossref_primary_10_1007_s10915_024_02584_4 crossref_primary_10_1007_s11425_022_2289_0 crossref_primary_10_1080_10556788_2022_2142583 |
| Cites_doi | 10.1109/TIP.2012.2190081 10.1109/TSP.2006.881263 10.1287/moor.2015.0722 10.1137/19M1304799 10.1214/09-AOS776 10.1007/s11222-008-9111-x 10.1214/15-AOS1388 10.1109/TIM.2016.2615449 10.1016/j.sigpro.2005.05.030 10.1016/j.acha.2014.10.001 10.1016/j.sigpro.2005.05.031 10.1109/TIP.2012.2205006 10.1137/080714488 10.1137/S0097539792240406 10.1137/11085476X 10.1007/s11425-016-9010-x 10.1007/s10107-016-0986-6 10.1137/100808071 10.1137/120869778 10.1016/j.acha.2019.04.003 10.1088/1361-6420/33/2/025010 10.1016/j.acha.2015.10.010 10.1016/j.acha.2018.02.002 10.1109/TIT.2012.2189196 10.1137/15M1028054 10.1023/A:1014813701864 10.1007/s10208-013-9161-0 10.1007/s11590-012-0456-x 10.1016/j.acha.2015.03.001 10.1137/18M1186009 10.1109/TSP.2016.2630028 10.1016/j.acha.2016.10.001 10.1007/978-3-642-02431-3 10.1109/ICASSP.2012.6288698 10.1109/CVPR.2012.6247852 |
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| Keywords | Sparse optimization 65F50 Local minimizer 90C46 Row selection problem Stationary point Multivariate regression 90C26 Global minimizer 90C27 |
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| References | Nikolova (CR24) 2016; 41 Esser, Moller, Osher, Sapiro, Xin (CR12) 2012; 21 Pan, Chen (CR28) 2021; 14 Natarajan (CR21) 1995; 24 CR11 CR33 Le (CR17) 2013; 7 Jiao, Jin, Lu (CR15) 2017; 65 Červinka, Kanzow, Schwartz (CR16) 2016; 160 Rockafellar, Wets (CR29) 1998 van den Berg, Friedlander (CR6) 2008; 31 Muduli, Mukherjee (CR20) 2017; 66 Satpathi, Chakraborty (CR31) 2017; 43 Tropp (CR34) 2006; 86 Obozinski, Taskar, Jordan (CR25) 2010; 20 Bauschke, Luke, Phan, Wang (CR1) 2014; 14 Yuan, Liu, Yan (CR37) 2012; 21 Chen, Huo (CR10) 2006; 54 Wen, Zhou, Liu, Lai, Tang (CR36) 2019; 47 Gillis (CR13) 2017; 25 Nikolova (CR23) 2013; 6 Beck, Eldar (CR2) 2013; 23 Beck, Hallak (CR3) 2015; 41 Blanchard, Leedy, Wu (CR8) 2020; 48 Lee, Bresler, Junge (CR18) 2012; 58 Bertsimas, King, Mazumder (CR5) 2016; 44 CR22 Lu, Zhang (CR19) 2013; 23 Zhang, Li (CR38) 2017; 33 Pan, Xiu, Fan (CR27) 2017; 60 Tropp, Gilbert, Strauss (CR35) 2006; 86 Bian, Chen (CR7) 2020; 58 Obozinski, Wainwright, Jordan (CR26) 2011; 39 Chen, Lu, Pong (CR9) 2016; 26 Jiao, Jin, Lu (CR14) 2015; 39 Belousov, Klatte (CR4) 2002; 22 Rockafellar (CR30) 2015 Shen, Xu, Zheng (CR32) 2016; 41 X Yuan (1124_CR37) 2012; 21 A Beck (1124_CR2) 2013; 23 M Nikolova (1124_CR23) 2013; 6 HH Bauschke (1124_CR1) 2014; 14 D Bertsimas (1124_CR5) 2016; 44 W Bian (1124_CR7) 2020; 58 JA Tropp (1124_CR34) 2006; 86 RT Rockafellar (1124_CR29) 1998 N Gillis (1124_CR13) 2017; 25 A Beck (1124_CR3) 2015; 41 1124_CR11 1124_CR33 Y Jiao (1124_CR14) 2015; 39 RT Rockafellar (1124_CR30) 2015 L Pan (1124_CR28) 2021; 14 J Chen (1124_CR10) 2006; 54 K Lee (1124_CR18) 2012; 58 G Obozinski (1124_CR26) 2011; 39 PR Muduli (1124_CR20) 2017; 66 M Červinka (1124_CR16) 2016; 160 JA Tropp (1124_CR35) 2006; 86 M Nikolova (1124_CR24) 2016; 41 L Pan (1124_CR27) 2017; 60 S Satpathi (1124_CR31) 2017; 43 E van den Berg (1124_CR6) 2008; 31 Z Lu (1124_CR19) 2013; 23 Y Jiao (1124_CR15) 2017; 65 1124_CR22 J Wen (1124_CR36) 2019; 47 B Natarajan (1124_CR21) 1995; 24 JD Blanchard (1124_CR8) 2020; 48 N Zhang (1124_CR38) 2017; 33 G Obozinski (1124_CR25) 2010; 20 HY Le (1124_CR17) 2013; 7 EG Belousov (1124_CR4) 2002; 22 E Esser (1124_CR12) 2012; 21 L Shen (1124_CR32) 2016; 41 X Chen (1124_CR9) 2016; 26 |
| References_xml | – ident: CR22 – volume: 21 start-page: 3239 year: 2012 end-page: 3252 ident: CR12 article-title: A convex model for nonnegative matrix factorization and dimensionality reduction on physical space publication-title: IEEE Trans. Image Process. doi: 10.1109/TIP.2012.2190081 – volume: 54 start-page: 4634 year: 2006 end-page: 4643 ident: CR10 article-title: Theoretical results on sparse representations of multiple-measurement vectors publication-title: IEEE Trans. Signal Process. doi: 10.1109/TSP.2006.881263 – volume: 41 start-page: 196 year: 2015 end-page: 223 ident: CR3 article-title: On the minimization over sparse symmetric sets: projections, optimality conditions, and algorithms publication-title: Math. Oper. Res. doi: 10.1287/moor.2015.0722 – volume: 14 start-page: 1 year: 2021 end-page: 25 ident: CR28 article-title: Group sparse optimization for image recovery using capped folded convace functions publication-title: SIAM J. Imaging Sci. doi: 10.1137/19M1304799 – volume: 39 start-page: 1 year: 2011 end-page: 47 ident: CR26 article-title: Support union recovery in high-dimensional multivariate regression publication-title: Ann. Stat. doi: 10.1214/09-AOS776 – volume: 20 start-page: 231 year: 2010 end-page: 252 ident: CR25 article-title: Joint covariate selection and joint subspace selection for multiple classification problems publication-title: Stat. Comput. doi: 10.1007/s11222-008-9111-x – volume: 44 start-page: 813 year: 2016 end-page: 852 ident: CR5 article-title: Best subset selection via a modern optimization lens publication-title: Ann. Stat. doi: 10.1214/15-AOS1388 – volume: 66 start-page: 234 year: 2017 end-page: 242 ident: CR20 article-title: A subspace projection-based joint sparse recovery method for structured biomedical signals publication-title: IEEE Trans. Instrum. Measur. doi: 10.1109/TIM.2016.2615449 – volume: 86 start-page: 572 issue: 3 year: 2006 end-page: 588 ident: CR35 article-title: Algorithms for simultaneous sparse approximation. Part I: greedy pursuit publication-title: Signal Process. doi: 10.1016/j.sigpro.2005.05.030 – ident: CR33 – volume: 25 start-page: 7 year: 2017 end-page: 16 ident: CR13 article-title: Introduction to nonnegative matrix factorization publication-title: SIAG-OPT Views News – volume: 39 start-page: 400 year: 2015 end-page: 426 ident: CR14 article-title: A primal dual active set with continuation algorithm for the -regularized optimization problem publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2014.10.001 – volume: 86 start-page: 589 year: 2006 end-page: 602 ident: CR34 article-title: Algorithms for simultaneous sparse approximation, Part II: convex relaxation publication-title: Signal Process. doi: 10.1016/j.sigpro.2005.05.031 – volume: 21 start-page: 4349 year: 2012 end-page: 4360 ident: CR37 article-title: Visual classification with multitask joint sparse representation publication-title: IEEE Trans. Image Process. doi: 10.1109/TIP.2012.2205006 – volume: 31 start-page: 890 year: 2008 end-page: 912 ident: CR6 article-title: Probing the Pareto frontier for basis pursuit solutions publication-title: SIAM J. Sci. Comput. doi: 10.1137/080714488 – volume: 24 start-page: 227 year: 1995 end-page: 234 ident: CR21 article-title: Sparse approximate solutions to linear systems publication-title: SIAM J. Comput. doi: 10.1137/S0097539792240406 – volume: 6 start-page: 904 year: 2013 end-page: 937 ident: CR23 article-title: Description of the minimizers of least squares regularized with -norm: uniqueness of the global minimizer publication-title: SIAM J. Imaging Sci. doi: 10.1137/11085476X – volume: 60 start-page: 759 year: 2017 end-page: 776 ident: CR27 article-title: Optimality conditions for sparse nonlinear programming publication-title: Sci. China Math. doi: 10.1007/s11425-016-9010-x – volume: 160 start-page: 353 year: 2016 end-page: 377 ident: CR16 article-title: Constraint qualifications and optimality conditions for optimization problems with cardinality constraints publication-title: Math. Program. doi: 10.1007/s10107-016-0986-6 – volume: 23 start-page: 2448 year: 2013 end-page: 2478 ident: CR19 article-title: Sparse approximation via penalty decomposition methods publication-title: SIAM J. Optim. doi: 10.1137/100808071 – volume: 23 start-page: 1480 year: 2013 end-page: 1509 ident: CR2 article-title: Sparsity constrained nonlinear optimization: optimality conditions and algorithms publication-title: SIAM J. Optim. doi: 10.1137/120869778 – volume: 48 start-page: 482 year: 2020 end-page: 495 ident: CR8 article-title: On rank awareness, thresholding, and MUSIC for joint sparse recovery publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2019.04.003 – volume: 33 start-page: 025010 year: 2017 ident: CR38 article-title: On optimal solutions of the constrained regularization and its penalty problem publication-title: Inverse Probl. doi: 10.1088/1361-6420/33/2/025010 – volume: 41 start-page: 237 year: 2016 end-page: 265 ident: CR24 article-title: Relationship between the optimal solutions of least squares regularized with -norm and constrained by -sparsity publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2015.10.010 – volume: 47 start-page: 948 year: 2019 end-page: 974 ident: CR36 article-title: Sharp sufficient conditions for stable recovery of block sparse signals by block orthogonal matching pursuit publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2018.02.002 – volume: 58 start-page: 3613 year: 2012 end-page: 3641 ident: CR18 article-title: Subspace methods for joint sparse recovery publication-title: IEEE Trans. Inform. Theory. doi: 10.1109/TIT.2012.2189196 – volume: 26 start-page: 1465 year: 2016 end-page: 1492 ident: CR9 article-title: Penalty methods for a class of non-Lipschitz optimization problems publication-title: SIAM J. Optim. doi: 10.1137/15M1028054 – volume: 22 start-page: 37 year: 2002 end-page: 48 ident: CR4 article-title: A Frank-Wolfe type theorem for convex polynomial programs publication-title: Comput. Optim. Appl. doi: 10.1023/A:1014813701864 – year: 2015 ident: CR30 publication-title: Convex Analysis – volume: 14 start-page: 63 year: 2014 end-page: 83 ident: CR1 article-title: Restricted normal cones and sparsity optimization with affine constraints publication-title: Found. Comput. Math. doi: 10.1007/s10208-013-9161-0 – volume: 7 start-page: 731 year: 2013 end-page: 743 ident: CR17 article-title: Generalized subdifferentials of the rank function publication-title: Optim. Lett. doi: 10.1007/s11590-012-0456-x – ident: CR11 – volume: 41 start-page: 26 year: 2016 end-page: 53 ident: CR32 article-title: Wavelet inpainting with the sparse regularization publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2015.03.001 – volume: 58 start-page: 858 year: 2020 end-page: 883 ident: CR7 article-title: A smoothing proximal gradient algorithm for nonsmooth convex regression with cardinality penalty publication-title: SIAM J. Numer. Anal. doi: 10.1137/18M1186009 – volume: 65 start-page: 998 year: 2017 end-page: 1012 ident: CR15 article-title: Group sparse recovery via the penalty: theory and algorithm publication-title: IEEE Trans. Signal Process. doi: 10.1109/TSP.2016.2630028 – volume: 43 start-page: 568 year: 2017 end-page: 576 ident: CR31 article-title: On the number of iterations for convergence of CoSaMP and Subspace Pursuit algorithms publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2016.10.001 – year: 1998 ident: CR29 publication-title: Variational Analysis doi: 10.1007/978-3-642-02431-3 – volume: 44 start-page: 813 year: 2016 ident: 1124_CR5 publication-title: Ann. Stat. doi: 10.1214/15-AOS1388 – volume: 58 start-page: 3613 year: 2012 ident: 1124_CR18 publication-title: IEEE Trans. Inform. Theory. doi: 10.1109/TIT.2012.2189196 – volume: 23 start-page: 1480 year: 2013 ident: 1124_CR2 publication-title: SIAM J. Optim. doi: 10.1137/120869778 – volume-title: Variational Analysis year: 1998 ident: 1124_CR29 doi: 10.1007/978-3-642-02431-3 – volume: 7 start-page: 731 year: 2013 ident: 1124_CR17 publication-title: Optim. Lett. doi: 10.1007/s11590-012-0456-x – volume: 66 start-page: 234 year: 2017 ident: 1124_CR20 publication-title: IEEE Trans. Instrum. Measur. doi: 10.1109/TIM.2016.2615449 – volume: 21 start-page: 4349 year: 2012 ident: 1124_CR37 publication-title: IEEE Trans. Image Process. doi: 10.1109/TIP.2012.2205006 – ident: 1124_CR33 doi: 10.1109/ICASSP.2012.6288698 – volume: 43 start-page: 568 year: 2017 ident: 1124_CR31 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2016.10.001 – volume: 48 start-page: 482 year: 2020 ident: 1124_CR8 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2019.04.003 – ident: 1124_CR11 doi: 10.1109/CVPR.2012.6247852 – volume: 41 start-page: 237 year: 2016 ident: 1124_CR24 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2015.10.010 – volume: 14 start-page: 1 year: 2021 ident: 1124_CR28 publication-title: SIAM J. Imaging Sci. doi: 10.1137/19M1304799 – volume: 31 start-page: 890 year: 2008 ident: 1124_CR6 publication-title: SIAM J. Sci. Comput. doi: 10.1137/080714488 – volume: 54 start-page: 4634 year: 2006 ident: 1124_CR10 publication-title: IEEE Trans. Signal Process. doi: 10.1109/TSP.2006.881263 – volume: 21 start-page: 3239 year: 2012 ident: 1124_CR12 publication-title: IEEE Trans. Image Process. doi: 10.1109/TIP.2012.2190081 – volume: 14 start-page: 63 year: 2014 ident: 1124_CR1 publication-title: Found. Comput. Math. doi: 10.1007/s10208-013-9161-0 – volume: 65 start-page: 998 year: 2017 ident: 1124_CR15 publication-title: IEEE Trans. Signal Process. doi: 10.1109/TSP.2016.2630028 – volume: 24 start-page: 227 year: 1995 ident: 1124_CR21 publication-title: SIAM J. Comput. doi: 10.1137/S0097539792240406 – volume: 33 start-page: 025010 year: 2017 ident: 1124_CR38 publication-title: Inverse Probl. doi: 10.1088/1361-6420/33/2/025010 – ident: 1124_CR22 – volume: 6 start-page: 904 year: 2013 ident: 1124_CR23 publication-title: SIAM J. Imaging Sci. doi: 10.1137/11085476X – volume: 26 start-page: 1465 year: 2016 ident: 1124_CR9 publication-title: SIAM J. Optim. doi: 10.1137/15M1028054 – volume: 60 start-page: 759 year: 2017 ident: 1124_CR27 publication-title: Sci. China Math. doi: 10.1007/s11425-016-9010-x – volume: 160 start-page: 353 year: 2016 ident: 1124_CR16 publication-title: Math. Program. doi: 10.1007/s10107-016-0986-6 – volume: 41 start-page: 196 year: 2015 ident: 1124_CR3 publication-title: Math. Oper. Res. doi: 10.1287/moor.2015.0722 – volume: 47 start-page: 948 year: 2019 ident: 1124_CR36 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2018.02.002 – volume: 25 start-page: 7 year: 2017 ident: 1124_CR13 publication-title: SIAG-OPT Views News – volume: 86 start-page: 589 year: 2006 ident: 1124_CR34 publication-title: Signal Process. doi: 10.1016/j.sigpro.2005.05.031 – volume: 20 start-page: 231 year: 2010 ident: 1124_CR25 publication-title: Stat. Comput. doi: 10.1007/s11222-008-9111-x – volume: 41 start-page: 26 year: 2016 ident: 1124_CR32 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2015.03.001 – volume: 22 start-page: 37 year: 2002 ident: 1124_CR4 publication-title: Comput. Optim. Appl. doi: 10.1023/A:1014813701864 – volume: 23 start-page: 2448 year: 2013 ident: 1124_CR19 publication-title: SIAM J. Optim. doi: 10.1137/100808071 – volume-title: Convex Analysis year: 2015 ident: 1124_CR30 – volume: 39 start-page: 1 year: 2011 ident: 1124_CR26 publication-title: Ann. Stat. doi: 10.1214/09-AOS776 – volume: 58 start-page: 858 year: 2020 ident: 1124_CR7 publication-title: SIAM J. Numer. Anal. doi: 10.1137/18M1186009 – volume: 39 start-page: 400 year: 2015 ident: 1124_CR14 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2014.10.001 – volume: 86 start-page: 572 issue: 3 year: 2006 ident: 1124_CR35 publication-title: Signal Process. doi: 10.1016/j.sigpro.2005.05.030 |
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| Title | Solution sets of three sparse optimization problems for multivariate regression |
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