Performance guarantees of transformed Schatten-1 regularization for exact low-rank matrix recovery
Low-rank matrix recovery aims to recover a matrix of minimum rank that subject to linear system constraint. It arises in various real world applications, such as recommender systems, image processing, and deep learning. Inspired by compressive sensing, the rank minimization can be relaxed to nuclear...
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| Published in | International journal of machine learning and cybernetics Vol. 12; no. 12; pp. 3379 - 3395 |
|---|---|
| Main Authors | , , , , , |
| Format | Journal Article |
| Language | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.12.2021
Springer Nature B.V |
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| Abstract | Low-rank matrix recovery aims to recover a matrix of minimum rank that subject to linear system constraint. It arises in various real world applications, such as recommender systems, image processing, and deep learning. Inspired by compressive sensing, the rank minimization can be relaxed to nuclear norm minimization. However, such a method treats all singular values of target matrix equally. To address this issue, recently the transformed Schatten-1 (TS1) penalty function was proposed and utilized to construct low-rank matrix recovery models. Unfortunately, the method for TS1-based models cannot provide both convergence accuracy and convergence speed. To alleviate such problems, this paper further investigates the basic properties of TS1 penalty function. And we describe a novel algorithm, which we called ATS1PGA, that is highly efficient in solving low-rank matrix recovery problems at a convergence rate of
O
(1/
N
), where
N
denotes the iterate count. In addition, we theoretically prove that the original rank minimization problem can be equivalently transformed into the TS1 optimization problem under certain conditions. Finally, extensive experimental results on real image data sets show that our proposed algorithm outperforms state-of-the-art methods in both accuracy and efficiency. In particular, our proposed algorithm is about 30 times faster than TS1 algorithm in solving low-rank matrix recovery problems. |
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| AbstractList | Low-rank matrix recovery aims to recover a matrix of minimum rank that subject to linear system constraint. It arises in various real world applications, such as recommender systems, image processing, and deep learning. Inspired by compressive sensing, the rank minimization can be relaxed to nuclear norm minimization. However, such a method treats all singular values of target matrix equally. To address this issue, recently the transformed Schatten-1 (TS1) penalty function was proposed and utilized to construct low-rank matrix recovery models. Unfortunately, the method for TS1-based models cannot provide both convergence accuracy and convergence speed. To alleviate such problems, this paper further investigates the basic properties of TS1 penalty function. And we describe a novel algorithm, which we called ATS1PGA, that is highly efficient in solving low-rank matrix recovery problems at a convergence rate of O(1/N), where N denotes the iterate count. In addition, we theoretically prove that the original rank minimization problem can be equivalently transformed into the TS1 optimization problem under certain conditions. Finally, extensive experimental results on real image data sets show that our proposed algorithm outperforms state-of-the-art methods in both accuracy and efficiency. In particular, our proposed algorithm is about 30 times faster than TS1 algorithm in solving low-rank matrix recovery problems. Low-rank matrix recovery aims to recover a matrix of minimum rank that subject to linear system constraint. It arises in various real world applications, such as recommender systems, image processing, and deep learning. Inspired by compressive sensing, the rank minimization can be relaxed to nuclear norm minimization. However, such a method treats all singular values of target matrix equally. To address this issue, recently the transformed Schatten-1 (TS1) penalty function was proposed and utilized to construct low-rank matrix recovery models. Unfortunately, the method for TS1-based models cannot provide both convergence accuracy and convergence speed. To alleviate such problems, this paper further investigates the basic properties of TS1 penalty function. And we describe a novel algorithm, which we called ATS1PGA, that is highly efficient in solving low-rank matrix recovery problems at a convergence rate of O (1/ N ), where N denotes the iterate count. In addition, we theoretically prove that the original rank minimization problem can be equivalently transformed into the TS1 optimization problem under certain conditions. Finally, extensive experimental results on real image data sets show that our proposed algorithm outperforms state-of-the-art methods in both accuracy and efficiency. In particular, our proposed algorithm is about 30 times faster than TS1 algorithm in solving low-rank matrix recovery problems. |
| Author | Wang, Wendong Chen, Wu Luo, Xiaohu Hu, Dong Wang, Zhi Wang, Jianjun |
| Author_xml | – sequence: 1 givenname: Zhi orcidid: 0000-0002-2167-830X surname: Wang fullname: Wang, Zhi email: chiw@swu.edu.cn organization: College of Computer and Information Science, Southwest University – sequence: 2 givenname: Dong surname: Hu fullname: Hu, Dong organization: College of Computer and Information Science, Southwest University – sequence: 3 givenname: Xiaohu surname: Luo fullname: Luo, Xiaohu organization: College of Computer and Information Science, Southwest University – sequence: 4 givenname: Wendong surname: Wang fullname: Wang, Wendong organization: College of Artificial Intelligence, Southwest University – sequence: 5 givenname: Jianjun surname: Wang fullname: Wang, Jianjun organization: School of Mathematics and Statistics, Southwest University – sequence: 6 givenname: Wu surname: Chen fullname: Chen, Wu email: chenwu@swu.edu.cn organization: College of Computer and Information Science, Southwest University |
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| Keywords | Nonconvex model Equivalence Transformed Schatten-1 penalty function Low-rank matrix recovery |
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| References | Gu S, Zhang L, Zuo W, Feng X (2014) Weighted nuclear norm minimization with application to image denoising. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2862–2869 ThuQGhanbariMScope of validity of PSNR in image/video quality assessmentElectron Lett2008441380080110.1049/el:20080522 Liu G, Liu Q, Yuan X (2017) A new theory for matrix completion. In: Proceedings of the advances in neural information processing systems, pp 785–794 Wang Z, Liu Y, Luo X, Wang J, Gao C, Peng D, Chen W (2021) Large-scale affine matrix rank minimization with a novel nonconvex regularizer. IEEE Trans Neural Netw Learn Syst (to be published) HalkoNMartinssonPTroppJFinding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositionsSIAM Rev2011532217288280663710.1137/090771806 WangZWangWWangJChenSFast and efficient algorithm for matrix completion via closed-form 2/3-thresholding operatorNeurocomputing201933021222210.1016/j.neucom.2018.10.065 LuoXZhouMLiSXiaYYouZZhuQLeungHIncorporation of efficient second-order solvers into latent factor models for accurate prediction of missing QoS dataIEEE Trans Cybern20184841216122810.1109/TCYB.2017.2685521 ParikhNBoydSProximal algorithmsFound Trends Optim20141312723910.1561/2400000003 MaSGoldfarbDChenLFixed point and Bregman iterative methods for matrix rank minimizationMath Progr20111281–2321353281096110.1007/s10107-009-0306-5 CuiAPengJLiHExact recovery low-rank matrix via transformed affine matrix rank minimizationNeurocomputing201831911210.1016/j.neucom.2018.05.092 RechtBFazelMParriloPGuaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimizationSIAM Rev2010523471501268054310.1137/070697835 ChenBYangZYangZAn algorithm for low-rank matrix factorization and its applicationsNeurocomputing20182751012102010.1016/j.neucom.2017.09.052 ZhangSXinJMinimization of transformed L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{1}$$\end{document} penalty: closed form representation and iterative thresholding algorithmsCommun Math Sci2017152511537362056710.4310/CMS.2017.v15.n2.a9 Fazel M (2002) Matrix rank minimization with applications. Ph.D. thesis, Stanford University Kang Z, Peng C, Cheng Q (2015) Robust PCA via nonconvex rank approximation. In: Proceedings of IEEE international conference on data mining, pp 211–220 LvJFanYA unified approach to model selection and sparse recovery using regularized least squaresAnn Stat20093763498352825495671369.62156 Yao Q, Kwok J (2015) Accelerated inexact soft-impute for fast large scale matrix completion. In: Proceedings of the international joint conference on artificial intelligence, pp 4002–4008 Li Q, Zhou Y, Liang Y, Varshney P (2017) Convergence analysis of proximal gradient with momentum for nonconvex optimization. In: Proceedings of the 34th international conference on machine learning, pp 2111–2119 OhTMatsushitaYTaiYKweonIFast randomized singular value thresholding for low-rank optimizationIEEE Trans Pattern Anal Mach Intell201840237639110.1109/TPAMI.2017.2677440 Schmidt M, Roux N, Bach F (2011) Convergence rates of inexact proximal gradient methods for convex optimization. In: Proceedings of the advances in neural information processing systems, pp 1458–1466 NesterovYA method for solving the convex programming problem with convergence rate O(1/k2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1/k^{2})$$\end{document}Dokl Akad Nauk SSSR1983272543547 WangZGaoCLuoXTangMWangJChenWAccelerated inexact matrix completion algorithm via closed-form q-thresholding (q=1/2,2/3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q=1/2,2/3)$$\end{document} operatorInt J Mach Learn Cybern2020112327233910.1007/s13042-020-01121-7 YaoQKwokJWangTLiuTLarge-scale low-rank matrix learning with nonconvex regularizersIEEE Trans Pattern Anal Mach Intell201941112628264310.1109/TPAMI.2018.2858249 TohK-CYunSAn accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problemsPac. J Optim20106361564027430471205.90218 Gu B, Huo Z, Huang H (2018) Inexact proximal gradient methods for non-convex and non-smooth optimization. In: Proceedings of the twenty-second AAAI conference on artificial intelligence, pp 3093–3100 ZhangSXinJMinimization of transformed L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{1}$$\end{document} penalty: theory, difference of convex function algorithm, and robust application in compressed sensingMath Progr20181691–230733610.1007/s10107-018-1236-x WangZLaiMLuZFanWDavulcuHYeJOrthogonal rank-one matrix pursuit for low rank matrix completionSIAM J Sci Comput2015371A488A514331383210.1137/130934271 CaiJ-FCandèsEJShenZA singular value thresholding algorithm for matrix completionSIAM J Optim201020419561982260024810.1137/080738970 Li H, Lin Z (2015) Accelerated proximal gradient methods for nonconvex programming. In: Proceedings of the advances in neural information processing systems, pp 379–387 FanJChowTDeep learning based matrix completionNeurocomputing201726679180310.1016/j.neucom.2017.05.074 PengDXiuNYuJS1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{1/2}$$\end{document} regularization methods and fixed point algorithms for affine rank minimization problemsComput Optim Appl201767543569365418510.1007/s10589-017-9898-5 CandèsEJWakinMBoydSEnhancing sparsity by reweighted l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{1}$$\end{document} minimizationJ Fourier Anal Appl200814877905246161110.1007/s00041-008-9045-x PengXZhangYTangHA unified framework for representation-based subspace clustering of out-of-sample and large-scale dataIEEE Trans Neural Netw Learn Syst2016271224992512357931310.1109/TNNLS.2015.2490080 LuoXZhouMLiSYouZXiaYZhuQA non-negative latent factor model for large-scale sparse matrices in recommender systems via alternating direction methodIEEE Trans Neural Netw Learn Syst2016273579592346565810.1109/TNNLS.2015.2415257 LiuGLiuQLiPLow-rank matrix completion in the presence of high coherenceIEEE Trans Signal Process2016642156235633354875610.1109/TSP.2016.2586753 LuCTangJYanSLinZNonconvex nonsmooth low rank minimization via iteratively reweighted nuclear normIEEE Trans Image Process2016251829839345544910.1109/TIP.2015.2511584 ZhaoFPengJCuiADesign strategy of thresholding operator for low-rank matrix recovery problemSignal Process202017111010.1016/j.sigpro.2020.107510 HuangCDingXFangCWenDRobust image restoration via adaptive low-rank approximation and joint kernel regressionIEEE Trans Image Process2014231252845297327507010.1109/TIP.2014.2363734 ZhangCNearly unbiased variable selection under minimax concave penaltyAnn Stat2010382894942260470110.1214/09-AOS729 FanJLiRVariable selection via nonconcal penalized likelihood and its oracle propertiesJ Am Stat Assoc2001964561348136010.1198/016214501753382273 Koren Y (2008) Factorization meets the neighborhood: a multifaceted collaborative filtering model. In: Proceedings of the 14th ACM SIGKDD international conference on knowledge discovery and data mining, pp 426–434 ZhangSYinPXinJTransformed Schatten-1 iterative thresholding algorithms for low rank matrix completionCommun Math Sci2017153839862362060610.4310/CMS.2017.v15.n3.a12 X Luo (1361_CR2) 2016; 27 1361_CR19 1361_CR18 F Zhao (1361_CR5) 2020; 171 1361_CR15 N Parikh (1361_CR32) 2014; 1 S Zhang (1361_CR30) 2018; 169 1361_CR11 1361_CR33 J-F Cai (1361_CR13) 2010; 20 1361_CR35 Z Wang (1361_CR36) 2015; 37 C Huang (1361_CR3) 2014; 23 1361_CR34 Q Yao (1361_CR25) 2019; 41 J Fan (1361_CR20) 2001; 96 EJ Candès (1361_CR22) 2008; 14 S Zhang (1361_CR26) 2017; 15 Z Wang (1361_CR16) 2019; 330 Z Wang (1361_CR17) 2020; 11 T Oh (1361_CR38) 2018; 40 D Peng (1361_CR23) 2017; 67 C Lu (1361_CR24) 2016; 25 A Cui (1361_CR31) 2018; 319 J Lv (1361_CR27) 2009; 37 S Zhang (1361_CR29) 2017; 15 X Luo (1361_CR6) 2018; 48 G Liu (1361_CR10) 2016; 64 1361_CR28 N Halko (1361_CR37) 2011; 53 B Recht (1361_CR12) 2010; 52 1361_CR40 1361_CR9 C Zhang (1361_CR21) 2010; 38 B Chen (1361_CR4) 2018; 275 S Ma (1361_CR41) 2011; 128 X Peng (1361_CR8) 2016; 27 Y Nesterov (1361_CR39) 1983; 27 K-C Toh (1361_CR14) 2010; 6 1361_CR1 Q Thu (1361_CR42) 2008; 44 J Fan (1361_CR7) 2017; 266 |
| References_xml | – reference: RechtBFazelMParriloPGuaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimizationSIAM Rev2010523471501268054310.1137/070697835 – reference: WangZWangWWangJChenSFast and efficient algorithm for matrix completion via closed-form 2/3-thresholding operatorNeurocomputing201933021222210.1016/j.neucom.2018.10.065 – reference: WangZGaoCLuoXTangMWangJChenWAccelerated inexact matrix completion algorithm via closed-form q-thresholding (q=1/2,2/3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q=1/2,2/3)$$\end{document} operatorInt J Mach Learn Cybern2020112327233910.1007/s13042-020-01121-7 – reference: Fazel M (2002) Matrix rank minimization with applications. Ph.D. thesis, Stanford University – reference: PengXZhangYTangHA unified framework for representation-based subspace clustering of out-of-sample and large-scale dataIEEE Trans Neural Netw Learn Syst2016271224992512357931310.1109/TNNLS.2015.2490080 – reference: ZhangSXinJMinimization of transformed L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{1}$$\end{document} penalty: theory, difference of convex function algorithm, and robust application in compressed sensingMath Progr20181691–230733610.1007/s10107-018-1236-x – reference: ChenBYangZYangZAn algorithm for low-rank matrix factorization and its applicationsNeurocomputing20182751012102010.1016/j.neucom.2017.09.052 – reference: TohK-CYunSAn accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problemsPac. J Optim20106361564027430471205.90218 – reference: Li Q, Zhou Y, Liang Y, Varshney P (2017) Convergence analysis of proximal gradient with momentum for nonconvex optimization. In: Proceedings of the 34th international conference on machine learning, pp 2111–2119 – reference: ZhaoFPengJCuiADesign strategy of thresholding operator for low-rank matrix recovery problemSignal Process202017111010.1016/j.sigpro.2020.107510 – reference: ParikhNBoydSProximal algorithmsFound Trends Optim20141312723910.1561/2400000003 – reference: LuCTangJYanSLinZNonconvex nonsmooth low rank minimization via iteratively reweighted nuclear normIEEE Trans Image Process2016251829839345544910.1109/TIP.2015.2511584 – reference: ThuQGhanbariMScope of validity of PSNR in image/video quality assessmentElectron Lett2008441380080110.1049/el:20080522 – reference: ZhangSXinJMinimization of transformed L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{1}$$\end{document} penalty: closed form representation and iterative thresholding algorithmsCommun Math Sci2017152511537362056710.4310/CMS.2017.v15.n2.a9 – reference: LuoXZhouMLiSXiaYYouZZhuQLeungHIncorporation of efficient second-order solvers into latent factor models for accurate prediction of missing QoS dataIEEE Trans Cybern20184841216122810.1109/TCYB.2017.2685521 – reference: LuoXZhouMLiSYouZXiaYZhuQA non-negative latent factor model for large-scale sparse matrices in recommender systems via alternating direction methodIEEE Trans Neural Netw Learn Syst2016273579592346565810.1109/TNNLS.2015.2415257 – reference: Gu B, Huo Z, Huang H (2018) Inexact proximal gradient methods for non-convex and non-smooth optimization. In: Proceedings of the twenty-second AAAI conference on artificial intelligence, pp 3093–3100 – reference: CaiJ-FCandèsEJShenZA singular value thresholding algorithm for matrix completionSIAM J Optim201020419561982260024810.1137/080738970 – reference: Kang Z, Peng C, Cheng Q (2015) Robust PCA via nonconvex rank approximation. In: Proceedings of IEEE international conference on data mining, pp 211–220 – reference: MaSGoldfarbDChenLFixed point and Bregman iterative methods for matrix rank minimizationMath Progr20111281–2321353281096110.1007/s10107-009-0306-5 – reference: LiuGLiuQLiPLow-rank matrix completion in the presence of high coherenceIEEE Trans Signal Process2016642156235633354875610.1109/TSP.2016.2586753 – reference: Li H, Lin Z (2015) Accelerated proximal gradient methods for nonconvex programming. In: Proceedings of the advances in neural information processing systems, pp 379–387 – reference: Yao Q, Kwok J (2015) Accelerated inexact soft-impute for fast large scale matrix completion. In: Proceedings of the international joint conference on artificial intelligence, pp 4002–4008 – reference: NesterovYA method for solving the convex programming problem with convergence rate O(1/k2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1/k^{2})$$\end{document}Dokl Akad Nauk SSSR1983272543547 – reference: Koren Y (2008) Factorization meets the neighborhood: a multifaceted collaborative filtering model. In: Proceedings of the 14th ACM SIGKDD international conference on knowledge discovery and data mining, pp 426–434 – reference: FanJLiRVariable selection via nonconcal penalized likelihood and its oracle propertiesJ Am Stat Assoc2001964561348136010.1198/016214501753382273 – reference: LvJFanYA unified approach to model selection and sparse recovery using regularized least squaresAnn Stat20093763498352825495671369.62156 – reference: YaoQKwokJWangTLiuTLarge-scale low-rank matrix learning with nonconvex regularizersIEEE Trans Pattern Anal Mach Intell201941112628264310.1109/TPAMI.2018.2858249 – reference: OhTMatsushitaYTaiYKweonIFast randomized singular value thresholding for low-rank optimizationIEEE Trans Pattern Anal Mach Intell201840237639110.1109/TPAMI.2017.2677440 – reference: CandèsEJWakinMBoydSEnhancing sparsity by reweighted l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{1}$$\end{document} minimizationJ Fourier Anal Appl200814877905246161110.1007/s00041-008-9045-x – reference: CuiAPengJLiHExact recovery low-rank matrix via transformed affine matrix rank minimizationNeurocomputing201831911210.1016/j.neucom.2018.05.092 – reference: ZhangCNearly unbiased variable selection under minimax concave penaltyAnn Stat2010382894942260470110.1214/09-AOS729 – reference: WangZLaiMLuZFanWDavulcuHYeJOrthogonal rank-one matrix pursuit for low rank matrix completionSIAM J Sci Comput2015371A488A514331383210.1137/130934271 – reference: PengDXiuNYuJS1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{1/2}$$\end{document} regularization methods and fixed point algorithms for affine rank minimization problemsComput Optim Appl201767543569365418510.1007/s10589-017-9898-5 – reference: HalkoNMartinssonPTroppJFinding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositionsSIAM Rev2011532217288280663710.1137/090771806 – reference: HuangCDingXFangCWenDRobust image restoration via adaptive low-rank approximation and joint kernel regressionIEEE Trans Image Process2014231252845297327507010.1109/TIP.2014.2363734 – reference: Wang Z, Liu Y, Luo X, Wang J, Gao C, Peng D, Chen W (2021) Large-scale affine matrix rank minimization with a novel nonconvex regularizer. IEEE Trans Neural Netw Learn Syst (to be published) – reference: ZhangSYinPXinJTransformed Schatten-1 iterative thresholding algorithms for low rank matrix completionCommun Math Sci2017153839862362060610.4310/CMS.2017.v15.n3.a12 – reference: Gu S, Zhang L, Zuo W, Feng X (2014) Weighted nuclear norm minimization with application to image denoising. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2862–2869 – reference: FanJChowTDeep learning based matrix completionNeurocomputing201726679180310.1016/j.neucom.2017.05.074 – reference: Schmidt M, Roux N, Bach F (2011) Convergence rates of inexact proximal gradient methods for convex optimization. In: Proceedings of the advances in neural information processing systems, pp 1458–1466 – reference: Liu G, Liu Q, Yuan X (2017) A new theory for matrix completion. In: Proceedings of the advances in neural information processing systems, pp 785–794 – volume: 67 start-page: 543 year: 2017 ident: 1361_CR23 publication-title: Comput Optim Appl doi: 10.1007/s10589-017-9898-5 – volume: 38 start-page: 894 issue: 2 year: 2010 ident: 1361_CR21 publication-title: Ann Stat doi: 10.1214/09-AOS729 – volume: 128 start-page: 321 issue: 1–2 year: 2011 ident: 1361_CR41 publication-title: Math Progr doi: 10.1007/s10107-009-0306-5 – volume: 53 start-page: 217 issue: 2 year: 2011 ident: 1361_CR37 publication-title: SIAM Rev doi: 10.1137/090771806 – volume: 14 start-page: 877 year: 2008 ident: 1361_CR22 publication-title: J Fourier Anal Appl doi: 10.1007/s00041-008-9045-x – volume: 20 start-page: 1956 issue: 4 year: 2010 ident: 1361_CR13 publication-title: SIAM J Optim doi: 10.1137/080738970 – volume: 171 start-page: 1 year: 2020 ident: 1361_CR5 publication-title: Signal Process doi: 10.1016/j.sigpro.2020.107510 – ident: 1361_CR11 – volume: 169 start-page: 307 issue: 1–2 year: 2018 ident: 1361_CR30 publication-title: Math Progr doi: 10.1007/s10107-018-1236-x – volume: 48 start-page: 1216 issue: 4 year: 2018 ident: 1361_CR6 publication-title: IEEE Trans Cybern doi: 10.1109/TCYB.2017.2685521 – volume: 1 start-page: 127 issue: 3 year: 2014 ident: 1361_CR32 publication-title: Found Trends Optim doi: 10.1561/2400000003 – ident: 1361_CR19 doi: 10.1109/CVPR.2014.366 – volume: 15 start-page: 511 issue: 2 year: 2017 ident: 1361_CR29 publication-title: Commun Math Sci doi: 10.4310/CMS.2017.v15.n2.a9 – volume: 37 start-page: 3498 issue: 6 year: 2009 ident: 1361_CR27 publication-title: Ann Stat – volume: 23 start-page: 5284 issue: 12 year: 2014 ident: 1361_CR3 publication-title: IEEE Trans Image Process doi: 10.1109/TIP.2014.2363734 – volume: 52 start-page: 471 issue: 3 year: 2010 ident: 1361_CR12 publication-title: SIAM Rev doi: 10.1137/070697835 – ident: 1361_CR15 – ident: 1361_CR40 – ident: 1361_CR1 doi: 10.1145/1401890.1401944 – volume: 275 start-page: 1012 year: 2018 ident: 1361_CR4 publication-title: Neurocomputing doi: 10.1016/j.neucom.2017.09.052 – volume: 27 start-page: 2499 issue: 12 year: 2016 ident: 1361_CR8 publication-title: IEEE Trans Neural Netw Learn Syst doi: 10.1109/TNNLS.2015.2490080 – volume: 6 start-page: 615 issue: 3 year: 2010 ident: 1361_CR14 publication-title: Pac. J Optim – volume: 96 start-page: 1348 issue: 456 year: 2001 ident: 1361_CR20 publication-title: J Am Stat Assoc doi: 10.1198/016214501753382273 – volume: 27 start-page: 543 issue: 2 year: 1983 ident: 1361_CR39 publication-title: Dokl Akad Nauk SSSR – volume: 25 start-page: 829 issue: 1 year: 2016 ident: 1361_CR24 publication-title: IEEE Trans Image Process doi: 10.1109/TIP.2015.2511584 – volume: 27 start-page: 579 issue: 3 year: 2016 ident: 1361_CR2 publication-title: IEEE Trans Neural Netw Learn Syst doi: 10.1109/TNNLS.2015.2415257 – volume: 319 start-page: 1 year: 2018 ident: 1361_CR31 publication-title: Neurocomputing doi: 10.1016/j.neucom.2018.05.092 – volume: 40 start-page: 376 issue: 2 year: 2018 ident: 1361_CR38 publication-title: IEEE Trans Pattern Anal Mach Intell doi: 10.1109/TPAMI.2017.2677440 – volume: 330 start-page: 212 year: 2019 ident: 1361_CR16 publication-title: Neurocomputing doi: 10.1016/j.neucom.2018.10.065 – ident: 1361_CR35 – volume: 41 start-page: 2628 issue: 11 year: 2019 ident: 1361_CR25 publication-title: IEEE Trans Pattern Anal Mach Intell doi: 10.1109/TPAMI.2018.2858249 – volume: 266 start-page: 791 year: 2017 ident: 1361_CR7 publication-title: Neurocomputing doi: 10.1016/j.neucom.2017.05.074 – volume: 11 start-page: 2327 year: 2020 ident: 1361_CR17 publication-title: Int J Mach Learn Cybern doi: 10.1007/s13042-020-01121-7 – ident: 1361_CR28 doi: 10.1109/ICDM.2015.15 – ident: 1361_CR33 – volume: 37 start-page: A488 issue: 1 year: 2015 ident: 1361_CR36 publication-title: SIAM J Sci Comput doi: 10.1137/130934271 – ident: 1361_CR9 – ident: 1361_CR34 doi: 10.1609/aaai.v32i1.11802 – volume: 15 start-page: 839 issue: 3 year: 2017 ident: 1361_CR26 publication-title: Commun Math Sci doi: 10.4310/CMS.2017.v15.n3.a12 – ident: 1361_CR18 doi: 10.1109/TNNLS.2021.3059711 – volume: 44 start-page: 800 issue: 13 year: 2008 ident: 1361_CR42 publication-title: Electron Lett doi: 10.1049/el:20080522 – volume: 64 start-page: 5623 issue: 21 year: 2016 ident: 1361_CR10 publication-title: IEEE Trans Signal Process doi: 10.1109/TSP.2016.2586753 |
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| Snippet | Low-rank matrix recovery aims to recover a matrix of minimum rank that subject to linear system constraint. It arises in various real world applications, such... |
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| SubjectTerms | Algorithms Artificial Intelligence Complex Systems Computational Intelligence Control Convergence Engineering Guarantees Image processing Mathematical analysis Mechatronics Optimization Original Article Pattern Recognition Penalty function Recommender systems Recovery Regularization Regularization methods Robotics Systems Biology |
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| Title | Performance guarantees of transformed Schatten-1 regularization for exact low-rank matrix recovery |
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