组稀疏模型及其算法综述
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摘要
稀疏性与组稀疏性在统计学、信号处理和机器学习等领域中具有重要的应用.本文总结和分析了不同组稀疏模型之间的区别与联系,比较了不同组稀疏模型的变量选择能力、变量组选择能力、变量选择一致性和变量组选择一致性,总结了组稀疏模型的各类求解算法并指出了各算法的优点和不足.最后,本文对组稀疏模型未来的研究方向进行了探讨.
The sparsity and group sparsity have important applications in the statistics,signal processing and machine learning. This paper summarized and analyzed the differences and relations between various group sparse models. In addition,we compared different models' variable selection ability,variable group selection ability,variable selection consistency and variable group selection consistency. We also summarized the algorithms of group sparse models and pointed the advantages and disadvantages of the algorithms. Finally,we point out the future research directions of the group sparse models.
The sparsity and group sparsity have important applications in the statistics,signal processing and machine learning. This paper summarized and analyzed the differences and relations between various group sparse models. In addition,we compared different models' variable selection ability,variable group selection ability,variable selection consistency and variable group selection consistency. We also summarized the algorithms of group sparse models and pointed the advantages and disadvantages of the algorithms. Finally,we point out the future research directions of the group sparse models.
引文
[1]Tibshirani R.Regression shrinkage and selection via the lasso[J].Journal of the Royal Statistical Society:Series B,1996,58(1):267-288.
[2]Yuan M,Lin Y.Model selection and estimation in regression w ith grouped variables[J].Journal of the Royal Statistical Society:Series B,2006,68(1):49-67.
[3]Turlach B A,Venables W N,Wright S J.Simultaneous variable selection[J].Technometrics,2005,47(3):349-363.
[4]Tropp J A.Algorithms for simultaneous sparse approximation[J].Signal Processing,2006,86(3):589-602.
[5]Quattoni A,Collins M,Darrell T.Transfer learning for image classification with sparse prototype representations[A].Proceedings of IEEE Conference on Computer Vision and Pattern Recognition[C].Alaska,USA:IEEE,2008.1-8.
[6]Schmidt M W,Murphy K P,Fung G,Rosales R.Structure learning in random fields for heart motion abnormality detection[A].Proceedings of IEEE Conference on Computer Vision and Pattern Recognition[C].Alaska,USA:IEEE,2008.1-8.
[7]Quattoni A,Carreras X,Collins M,Darrell T.An efficient projection for L1,∞regularization[A].Proceedings of the26th Annual International Conference on M achine Learning[C].Quebec,Canada:ACM,2009.857-864.
[8]Vogt J E,Roth V.The group-Lasso:l1,∞regularization versus l1,2regularization[A].Proceedings of the 32nd DAGM conference on Pattern recognition[C].Darmstadt,Germany:Springer-Verlag,2010.252-261.
[9]Huang J,Zhang T.The benefit of group sparsity[J].The Annals of Statistics,2010,38(4):1978-2004.
[10]Sra S.Fast projections onto1,q-norm balls for grouped feature selection[J].Lecture Notes in Computer Science,2011:305-317.
[11]Kowalski M.Sparse regression using mixed norms[J].Applied and Computational Harmonic Analysis,2009,27(3):303-324.
[12]Rakotomamonjy A,et al.Lp-Lq penalty for sparse linear and sparse multiple kernel multi-task learning[J].IEEE Transactions on Neural Networks,2011,22(8):1307-1320.
[13]Simon N,Tibshirani R.Standardization and the group lasso penalty[J].Statistica Sinica,2012,22(3):983-1001.
[14]Bunea F,Lederer J,She Y.The group square-root lasso:theoretical properties and fast algorithms[J].IEEE Transactions on Information Theory,2014,60(2):1313-1325.
[15]Belloni A,Chernozhukov V,Wang L.Square-root lasso:pivotal recovery of sparse signals via conic programming[J].Biometrika,2011,98(4):791-806.
[16]Wang H,Leng C.A note on adaptive group lasso[J].Computational Statistics and Data Analysis,2008,52(12):5277-5286.
[17]Wei F,Huang J.Consistent group selection in high-dimensional linear regression[J].Bernoulli,2010,16(4):1369-1384.
[18]Zou H.The adaptive lasso and its oracle properties[J].Journal of the American statistical association,2006,101(476):1418-1429.
[19]Zhang H H,Lu W.Adaptive lasso for Cox's proportional hazards model[J].Biometrika,2007,94(3):691-703.
[20]Huang J,Ma S,Zhang C H.Adaptive lasso for sparse high-dimensional regression models[J].Statistica Sinica,2008,18(4):1603-1618.
[21]Simon N,et al.A sparse-group lasso[J].Journal of Computational and Graphical Statistics,2013,22(2):231-245.
[22]Chatterjee S,Steinhaeuser K,Banerjee A,et al.Sparse group lasso:consistency and climate applications[A].Proceedings of the 12th SIAM International Conference on Data M ining[C].California,USA:Omnipress,2012.47-58.
[23]Zhu X,Huang Z,et al.Video-to-shot tag allocation by w eighted sparse group lasso[A].Proceedings of the 19th ACM International Conference on M ultimedia[C].Scottsdale:ACM,2011.1501-1504.
[24]Raman S,Fuchs T J,Wild P J,Dahl E,Roth V.The Bayesian group-Lasso for analyzing contingency tables[A].Proceedings of the 26th Annual International Conference on M achine Learning[C].M ontreal,Canada:ACM,2009.881-888.
[25]Chandran M.Analysis of Bayesian group-Lasso in regression models[D].Florida,USA:University of Florida,2011.
[26]Meier L,Van De Geer S,Bühlmann P.The group lasso for logistic regression[J].Journal of the Royal Statistical Society:Series B,2008,70(1):53-71.
[27]Kim Y,Kim J,et al.Blockwise sparse regression[J].Statistica Sinica,2006,16(2):375-390.
[28]Sun H,Wang S.Penalized logistic regression for high-dimensional DNA methylation data w ith case-control studies[J].Bioinformatics,2012,28(10):1368-1375.
[29]Wu F,Yuan Y,Zhuang Y.Heterogeneous feature selection by group lasso w ith logistic regression[A].Proceedings of the International Conference on M ultimedia[C].Firenze,Italy:ACM,2010.983-986.
[30]Wang S,et al.Hierarchically penalized Cox regression with grouped variables[J].Biometrika,2009,96(2):307-322.
[31]Liu X,Wang Z,Wu Y.Group variable selection and estimation in the tobit censored response model[J].Computational Statistics and Data Analysis,2013,60:80-89.
[32]Ji Y,Lin N,Zhang B.Model selection in binary and tobit quantile regression using the Gibbs sampler[J].Computational Statistics&Data Analysis,2012,56(4):827-839.
[33]Yin J,Chen X,Xing E P.Group sparse additive models[A].Proceedings of the 29th International Conference on M achine Learning[C].Scotland,UK:Omnipress,2012.871-878.
[34]Jiang D F.Concave selection in Generalized linear models[D].Iowa,USA:University of Iowa,2012.
[35]Wang L,Chen G,Li H.Group SCAD regression analysis for microarray time course gene expression data[J].Bioinformatics,2007,23(12):1486-1494.
[36]Fan J,Li R.Variable selection via nonconcave penalized likelihood and its oracle properties[J].Journal of the American Statistical Association,2001,96(456):1348-1360.
[37]Breheny P,Huang J.Penalized methods for bi-level variable selection[J].Statistics and Its Interface,2009,2(3):369-380.
[38]Huang J,Breheny P,Ma S.A selective review of group selection in high-dimensional models[J].Statistical Science,2012,27(4):481-499.
[39]Zhang C H.Nearly unbiased variable selection under minimax concave penalty[J].Annals of Statistics,2010,38(2):894-942.
[40]Huang J,Ma S,Xie H,Zhang C H.A group bridge approach for variable selection[J].Biometrika,2009,96(2):339-355.
[41]Ma S,Huang J,Song X.Integrative analysis and variable selection w ith multiple high-dimensional data sets[J].Biostatistics,2011,12(4):763-775.
[42]Ma S,Dai Y,Huang J,et al.Identification of breast cancer prognosis markers via integrative analysis[J].Computational Statistics&Data Analysis,2012,56(9):2718-2728.
[43]Seetharaman I.Consistent bi-level variable selection via composite group bridge penalized regression[D].Kansas,USA:Kansas State Univesity,2013.
[44]Fu W J.Penalized regressions:the bridge versus the Lasso[J].Journal of Computational and Graphical Statistics,1998,7(3):397-416.
[45]Jenatton R,Audibert J Y,Bach F.Structured variable selection w ith sparsity-inducing norms[J].The Journal of M achine Learning Research,2011,12:2777-2824.
[46]Mosci S,et al.A primal-dual algorithm for group sparse regularization with overlapping groups[A].Proceedings of Advances in Neural Information Processing Systems 23:24th Annual Conference on Neural Information Processing Systems[C].Canada:Curran Associates,2010.2604-2612.
[47]Percival D.Theoretical properties of the overlapping groups Lasso[J].Electronic Journal of Statistics,2012,6:269-288.
[48]Yuan L,Liu J,Ye J.Efficient methods for overlapping group lasso[J].IEEE Transactions on Pattern Analysis and M achine Intelligence,2013,35(9):2104-2116.
[49]Percival D.Theoretical properties of the overlapping groups lasso[J].Electronic Journal of Statistics,2012,6:269-288.
[50]Jenatton R,Mairal J,Obozinski G,Bach F.Proximal methods for hierarchical sparse coding[J].Journal of M achine Learning Research,2011,12:2297-2334.
[51]Liu J,Ye J P.Moreau-Yosida regularization for grouped tree structure learning[A].Proceedings of Advances in Neural Information Processing Systems 23:24th Annual Conference on Neural Information Processing Systems[C].Vancouver,Canada:Curran Associates,2010.1459-1467.
[52]Martins A F T,Smith N A,et al.Structured sparsity in structured prediction[A].Proceedings of the 2010 Conference on Empirical M ethods in Natural Language Processing[C].M assachusetts,America:ACL,2011.1500-1511.
[53]Zhao P,Rocha G,Yu B.Grouped and hierarchical model selection through composite absolute penalties[J].The Annals of Statistics,2009,(6):3468-3497.
[54]Kim S,Xing E P.Tree-guided group Lasso for multi-task regression w ith structured sparsity[A].Proceedings of the27th International Conference on M achine Learning[C].Haifa,Israel:Omnipress,2010.543-550.
[55]Kim S,Xing E P.Tree-guided group Lasso for multi-response regression w ith structured sparsity w ith an application to e QTL mapping[J].The Annals of Applied Statistics,2012,6(3):1095-1117.
[56]Zhao P,Yu B.On model selection consistency of lasso[J].The Journal of Machine Learning Research,2006,7:2541-2563.
[57]Bach F R.Consistency of the group lasso and multiple kernel learning[J].The Journal of M achine Learning Research,2008,9:1179-1225.
[58]Zhang C H,Huang J.The sparsity and bias of the lasso selection in high-dimensional linear regression[J].The Annals of Statistics,2008,36(4):1567-1594.
[59]Wei F,Huang J.Consistent group selection in high-dimensional linear regression[J].Bernoulli,2010,16(4):1369-1384.
[60]Bickel P J,Ritov Y,Tsybakov A B.Simultaneous analysis of lasso and Dantzig selector[J].The Annals of Statistics,2009,37(4):1705-1732.
[61]Lounici K,Pontil M,Van D G,Tsybakov A B.Oracle inequalities and optimal inference under group sparsity[J].The Annals of Statistics,2011,39(4),2164-2204.
[62]Nesterov Y.Smooth minimization of non-smooth functions[J].M athematical Programming,2005,103(1):127-152.
[63]Zou H,Li R.One-step sparse estimates in nonconcave penalized likelihood models[J].Annals of statistics,2008,36(4):1509-1533.
[64]Yang Y,Zou H.A fast unified algorithm for solving group-lasso penalized learning problems[J].Journal of Computational and Graphical Statistics,2012:328-361.
[65]Qin Z,Scheinberg K,Goldfarb D.Efficient block-coordinate descent algorithms for the group lasso[J].M athematical Programming Computation,2013:143-169.
[66]Tseng P,Yun S.A coordinate gradient descent method for nonsmooth separable minimization[J].M athematical Programming,2009,117:387-423.
[67]Roth V,Fischer B.The group-lasso for generalized linear models:uniqueness of solutions and efficient algorithms[A].Proceedings of The 25th International Conference on M achine learning[C].Helsinki,Finland:ACM,2008.848-855.
[68]Boyd S,Parikh N,Chu E,et al.Distributed optimization and statistical learning via the alternating direction method of multipliers[J].Foundations and Trends in M achine Learning,2011,3(1):1-122.
[69]Efron B,Hastie T,Johnstone I,et al.Least angle regression[J].The Annals of statistics,2004,32(2):407-499.
[70]Meinshausen N,et al.Stability selection[J].Journal of the Royal Statistical Society:Series B,2010,72(4):417-473.
[71]Choon C L.Minimax concave bridge penalty function for variable selection[D].Singapore:National University of Singapore,2012.
[72]Mazumder R,Friedman J H,Hastie T.Sparse Net:coordinate descent w ith nonconvex penalties[J].Journal of the American Statistical Association,2011,106(495):1125-1138.
[73]Kwon S,Kim Y,Choi H.Sparse bridge estimation with a diverging number of parameters[J].Statistics and Its Interface,2012,6:231-242.
[74]Candes E J,Wakin M B,Boyd S P.Enhancing sparsity by rew eighted L1minimization[J].Journal of Fourier Analysis and Applications,2008,14(5-6):877-905.
[75]Van D G,Bühlmann P.On the conditions used to prove oracle results for the lasso[J].Electronic Journal of Statistics,2009,3:1360-1392.
[76]Zhang T.Some sharp performance bounds for least squares regression w ith L1regularization[J].The Annals of Statistics,2009,37(5A):2109-2144.
[77]Ye F,Zhang C H.Rate minimaxity of the lasso and Dantzig selector for the Lq loss in Lr balls[J].The Journal of M achine Learning Research,2010,11:3519-3540.
[2]Yuan M,Lin Y.Model selection and estimation in regression w ith grouped variables[J].Journal of the Royal Statistical Society:Series B,2006,68(1):49-67.
[3]Turlach B A,Venables W N,Wright S J.Simultaneous variable selection[J].Technometrics,2005,47(3):349-363.
[4]Tropp J A.Algorithms for simultaneous sparse approximation[J].Signal Processing,2006,86(3):589-602.
[5]Quattoni A,Collins M,Darrell T.Transfer learning for image classification with sparse prototype representations[A].Proceedings of IEEE Conference on Computer Vision and Pattern Recognition[C].Alaska,USA:IEEE,2008.1-8.
[6]Schmidt M W,Murphy K P,Fung G,Rosales R.Structure learning in random fields for heart motion abnormality detection[A].Proceedings of IEEE Conference on Computer Vision and Pattern Recognition[C].Alaska,USA:IEEE,2008.1-8.
[7]Quattoni A,Carreras X,Collins M,Darrell T.An efficient projection for L1,∞regularization[A].Proceedings of the26th Annual International Conference on M achine Learning[C].Quebec,Canada:ACM,2009.857-864.
[8]Vogt J E,Roth V.The group-Lasso:l1,∞regularization versus l1,2regularization[A].Proceedings of the 32nd DAGM conference on Pattern recognition[C].Darmstadt,Germany:Springer-Verlag,2010.252-261.
[9]Huang J,Zhang T.The benefit of group sparsity[J].The Annals of Statistics,2010,38(4):1978-2004.
[10]Sra S.Fast projections onto1,q-norm balls for grouped feature selection[J].Lecture Notes in Computer Science,2011:305-317.
[11]Kowalski M.Sparse regression using mixed norms[J].Applied and Computational Harmonic Analysis,2009,27(3):303-324.
[12]Rakotomamonjy A,et al.Lp-Lq penalty for sparse linear and sparse multiple kernel multi-task learning[J].IEEE Transactions on Neural Networks,2011,22(8):1307-1320.
[13]Simon N,Tibshirani R.Standardization and the group lasso penalty[J].Statistica Sinica,2012,22(3):983-1001.
[14]Bunea F,Lederer J,She Y.The group square-root lasso:theoretical properties and fast algorithms[J].IEEE Transactions on Information Theory,2014,60(2):1313-1325.
[15]Belloni A,Chernozhukov V,Wang L.Square-root lasso:pivotal recovery of sparse signals via conic programming[J].Biometrika,2011,98(4):791-806.
[16]Wang H,Leng C.A note on adaptive group lasso[J].Computational Statistics and Data Analysis,2008,52(12):5277-5286.
[17]Wei F,Huang J.Consistent group selection in high-dimensional linear regression[J].Bernoulli,2010,16(4):1369-1384.
[18]Zou H.The adaptive lasso and its oracle properties[J].Journal of the American statistical association,2006,101(476):1418-1429.
[19]Zhang H H,Lu W.Adaptive lasso for Cox's proportional hazards model[J].Biometrika,2007,94(3):691-703.
[20]Huang J,Ma S,Zhang C H.Adaptive lasso for sparse high-dimensional regression models[J].Statistica Sinica,2008,18(4):1603-1618.
[21]Simon N,et al.A sparse-group lasso[J].Journal of Computational and Graphical Statistics,2013,22(2):231-245.
[22]Chatterjee S,Steinhaeuser K,Banerjee A,et al.Sparse group lasso:consistency and climate applications[A].Proceedings of the 12th SIAM International Conference on Data M ining[C].California,USA:Omnipress,2012.47-58.
[23]Zhu X,Huang Z,et al.Video-to-shot tag allocation by w eighted sparse group lasso[A].Proceedings of the 19th ACM International Conference on M ultimedia[C].Scottsdale:ACM,2011.1501-1504.
[24]Raman S,Fuchs T J,Wild P J,Dahl E,Roth V.The Bayesian group-Lasso for analyzing contingency tables[A].Proceedings of the 26th Annual International Conference on M achine Learning[C].M ontreal,Canada:ACM,2009.881-888.
[25]Chandran M.Analysis of Bayesian group-Lasso in regression models[D].Florida,USA:University of Florida,2011.
[26]Meier L,Van De Geer S,Bühlmann P.The group lasso for logistic regression[J].Journal of the Royal Statistical Society:Series B,2008,70(1):53-71.
[27]Kim Y,Kim J,et al.Blockwise sparse regression[J].Statistica Sinica,2006,16(2):375-390.
[28]Sun H,Wang S.Penalized logistic regression for high-dimensional DNA methylation data w ith case-control studies[J].Bioinformatics,2012,28(10):1368-1375.
[29]Wu F,Yuan Y,Zhuang Y.Heterogeneous feature selection by group lasso w ith logistic regression[A].Proceedings of the International Conference on M ultimedia[C].Firenze,Italy:ACM,2010.983-986.
[30]Wang S,et al.Hierarchically penalized Cox regression with grouped variables[J].Biometrika,2009,96(2):307-322.
[31]Liu X,Wang Z,Wu Y.Group variable selection and estimation in the tobit censored response model[J].Computational Statistics and Data Analysis,2013,60:80-89.
[32]Ji Y,Lin N,Zhang B.Model selection in binary and tobit quantile regression using the Gibbs sampler[J].Computational Statistics&Data Analysis,2012,56(4):827-839.
[33]Yin J,Chen X,Xing E P.Group sparse additive models[A].Proceedings of the 29th International Conference on M achine Learning[C].Scotland,UK:Omnipress,2012.871-878.
[34]Jiang D F.Concave selection in Generalized linear models[D].Iowa,USA:University of Iowa,2012.
[35]Wang L,Chen G,Li H.Group SCAD regression analysis for microarray time course gene expression data[J].Bioinformatics,2007,23(12):1486-1494.
[36]Fan J,Li R.Variable selection via nonconcave penalized likelihood and its oracle properties[J].Journal of the American Statistical Association,2001,96(456):1348-1360.
[37]Breheny P,Huang J.Penalized methods for bi-level variable selection[J].Statistics and Its Interface,2009,2(3):369-380.
[38]Huang J,Breheny P,Ma S.A selective review of group selection in high-dimensional models[J].Statistical Science,2012,27(4):481-499.
[39]Zhang C H.Nearly unbiased variable selection under minimax concave penalty[J].Annals of Statistics,2010,38(2):894-942.
[40]Huang J,Ma S,Xie H,Zhang C H.A group bridge approach for variable selection[J].Biometrika,2009,96(2):339-355.
[41]Ma S,Huang J,Song X.Integrative analysis and variable selection w ith multiple high-dimensional data sets[J].Biostatistics,2011,12(4):763-775.
[42]Ma S,Dai Y,Huang J,et al.Identification of breast cancer prognosis markers via integrative analysis[J].Computational Statistics&Data Analysis,2012,56(9):2718-2728.
[43]Seetharaman I.Consistent bi-level variable selection via composite group bridge penalized regression[D].Kansas,USA:Kansas State Univesity,2013.
[44]Fu W J.Penalized regressions:the bridge versus the Lasso[J].Journal of Computational and Graphical Statistics,1998,7(3):397-416.
[45]Jenatton R,Audibert J Y,Bach F.Structured variable selection w ith sparsity-inducing norms[J].The Journal of M achine Learning Research,2011,12:2777-2824.
[46]Mosci S,et al.A primal-dual algorithm for group sparse regularization with overlapping groups[A].Proceedings of Advances in Neural Information Processing Systems 23:24th Annual Conference on Neural Information Processing Systems[C].Canada:Curran Associates,2010.2604-2612.
[47]Percival D.Theoretical properties of the overlapping groups Lasso[J].Electronic Journal of Statistics,2012,6:269-288.
[48]Yuan L,Liu J,Ye J.Efficient methods for overlapping group lasso[J].IEEE Transactions on Pattern Analysis and M achine Intelligence,2013,35(9):2104-2116.
[49]Percival D.Theoretical properties of the overlapping groups lasso[J].Electronic Journal of Statistics,2012,6:269-288.
[50]Jenatton R,Mairal J,Obozinski G,Bach F.Proximal methods for hierarchical sparse coding[J].Journal of M achine Learning Research,2011,12:2297-2334.
[51]Liu J,Ye J P.Moreau-Yosida regularization for grouped tree structure learning[A].Proceedings of Advances in Neural Information Processing Systems 23:24th Annual Conference on Neural Information Processing Systems[C].Vancouver,Canada:Curran Associates,2010.1459-1467.
[52]Martins A F T,Smith N A,et al.Structured sparsity in structured prediction[A].Proceedings of the 2010 Conference on Empirical M ethods in Natural Language Processing[C].M assachusetts,America:ACL,2011.1500-1511.
[53]Zhao P,Rocha G,Yu B.Grouped and hierarchical model selection through composite absolute penalties[J].The Annals of Statistics,2009,(6):3468-3497.
[54]Kim S,Xing E P.Tree-guided group Lasso for multi-task regression w ith structured sparsity[A].Proceedings of the27th International Conference on M achine Learning[C].Haifa,Israel:Omnipress,2010.543-550.
[55]Kim S,Xing E P.Tree-guided group Lasso for multi-response regression w ith structured sparsity w ith an application to e QTL mapping[J].The Annals of Applied Statistics,2012,6(3):1095-1117.
[56]Zhao P,Yu B.On model selection consistency of lasso[J].The Journal of Machine Learning Research,2006,7:2541-2563.
[57]Bach F R.Consistency of the group lasso and multiple kernel learning[J].The Journal of M achine Learning Research,2008,9:1179-1225.
[58]Zhang C H,Huang J.The sparsity and bias of the lasso selection in high-dimensional linear regression[J].The Annals of Statistics,2008,36(4):1567-1594.
[59]Wei F,Huang J.Consistent group selection in high-dimensional linear regression[J].Bernoulli,2010,16(4):1369-1384.
[60]Bickel P J,Ritov Y,Tsybakov A B.Simultaneous analysis of lasso and Dantzig selector[J].The Annals of Statistics,2009,37(4):1705-1732.
[61]Lounici K,Pontil M,Van D G,Tsybakov A B.Oracle inequalities and optimal inference under group sparsity[J].The Annals of Statistics,2011,39(4),2164-2204.
[62]Nesterov Y.Smooth minimization of non-smooth functions[J].M athematical Programming,2005,103(1):127-152.
[63]Zou H,Li R.One-step sparse estimates in nonconcave penalized likelihood models[J].Annals of statistics,2008,36(4):1509-1533.
[64]Yang Y,Zou H.A fast unified algorithm for solving group-lasso penalized learning problems[J].Journal of Computational and Graphical Statistics,2012:328-361.
[65]Qin Z,Scheinberg K,Goldfarb D.Efficient block-coordinate descent algorithms for the group lasso[J].M athematical Programming Computation,2013:143-169.
[66]Tseng P,Yun S.A coordinate gradient descent method for nonsmooth separable minimization[J].M athematical Programming,2009,117:387-423.
[67]Roth V,Fischer B.The group-lasso for generalized linear models:uniqueness of solutions and efficient algorithms[A].Proceedings of The 25th International Conference on M achine learning[C].Helsinki,Finland:ACM,2008.848-855.
[68]Boyd S,Parikh N,Chu E,et al.Distributed optimization and statistical learning via the alternating direction method of multipliers[J].Foundations and Trends in M achine Learning,2011,3(1):1-122.
[69]Efron B,Hastie T,Johnstone I,et al.Least angle regression[J].The Annals of statistics,2004,32(2):407-499.
[70]Meinshausen N,et al.Stability selection[J].Journal of the Royal Statistical Society:Series B,2010,72(4):417-473.
[71]Choon C L.Minimax concave bridge penalty function for variable selection[D].Singapore:National University of Singapore,2012.
[72]Mazumder R,Friedman J H,Hastie T.Sparse Net:coordinate descent w ith nonconvex penalties[J].Journal of the American Statistical Association,2011,106(495):1125-1138.
[73]Kwon S,Kim Y,Choi H.Sparse bridge estimation with a diverging number of parameters[J].Statistics and Its Interface,2012,6:231-242.
[74]Candes E J,Wakin M B,Boyd S P.Enhancing sparsity by rew eighted L1minimization[J].Journal of Fourier Analysis and Applications,2008,14(5-6):877-905.
[75]Van D G,Bühlmann P.On the conditions used to prove oracle results for the lasso[J].Electronic Journal of Statistics,2009,3:1360-1392.
[76]Zhang T.Some sharp performance bounds for least squares regression w ith L1regularization[J].The Annals of Statistics,2009,37(5A):2109-2144.
[77]Ye F,Zhang C H.Rate minimaxity of the lasso and Dantzig selector for the Lq loss in Lr balls[J].The Journal of M achine Learning Research,2010,11:3519-3540.