{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T23:14:56Z","timestamp":1787008496165,"version":"build-2736575974"},"reference-count":52,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2019,11,23]],"date-time":"2019-11-23T00:00:00Z","timestamp":1574467200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,3,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>We develop a Bayesian methodology aimed at simultaneously estimating low-rank and row-sparse matrices in a high-dimensional multiple-response linear regression model. We consider a carefully devised shrinkage prior on the matrix of regression coefficients which obviates the need to specify a prior on the rank, and shrinks the regression matrix towards low-rank and row-sparse structures. We provide theoretical support to the proposed methodology by proving minimax optimality of the posterior mean under the prediction risk in ultra-high-dimensional settings where the number of predictors can grow subexponentially relative to the sample size. A one-step post-processing scheme induced by group lasso penalties on the rows of the estimated coefficient matrix is proposed for variable selection, with default choices of tuning parameters. We additionally provide an estimate of the rank using a novel optimization function achieving dimension reduction in the covariate space. We exhibit the performance of the proposed methodology in an extensive simulation study and a real data example.<\/jats:p>","DOI":"10.1093\/biomet\/asz056","type":"journal-article","created":{"date-parts":[[2019,11,20]],"date-time":"2019-11-20T07:13:09Z","timestamp":1574233989000},"page":"205-221","source":"Crossref","is-referenced-by-count":19,"title":["Bayesian sparse multiple regression for simultaneous rank reduction and variable selection"],"prefix":"10.1093","volume":"107","author":[{"given":"Antik","family":"Chakraborty","sequence":"first","affiliation":[{"name":"Department of Statistics, Texas A&M University, College Station, Texas 77843, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anirban","family":"Bhattacharya","sequence":"additional","affiliation":[{"name":"Department of Statistics, Texas A&M University, College Station, Texas 77843, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bani K","family":"Mallick","sequence":"additional","affiliation":[{"name":"Department of Statistics, Texas A&M University, College Station, Texas 77843, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2019,11,23]]},"reference":[{"key":"2020021310531777800_B1","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/978-3-642-40935-6_22","volume-title":"Proc. 24th Int. Conf. Algorithmic Learning Theory","author":"Alquier,","year":"2013"},{"key":"2020021310531777800_B2","volume-title":"Multivariate Statistical Analysis","author":"Anderson,","year":"1984"},{"key":"2020021310531777800_B3","first-page":"193","article-title":"Specification and misspecification in reduced rank regression","volume":"64","author":"Anderson,","year":"2002","journal-title":"Sankhy\u0101"},{"key":"2020021310531777800_B4","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1214\/aoms\/1177729580","article-title":"Estimating linear restrictions on regression coefficients for multivariate normal distributions","volume":"22","author":"Anderson,","year":"1951","journal-title":"Ann. Math. Statist."},{"key":"2020021310531777800_B5","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1093\/biomet\/ast028","article-title":"Posterior consistency in linear models under shrinkage priors","volume":"100","author":"Armagan,","year":"2013","journal-title":"Biometrika"},{"key":"2020021310531777800_B6","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1109\/TIP.2010.2080278","article-title":"Variational Bayesian super resolution","volume":"20","author":"Babacan,","year":"2011","journal-title":"IEEE Trans. Image Proces."},{"key":"2020021310531777800_B7","doi-asserted-by":"crossref","first-page":"870","DOI":"10.1214\/009053604000000238","article-title":"Optimal predictive model selection","volume":"32","author":"Barbieri,","year":"2004","journal-title":"Ann. Statist."},{"key":"2020021310531777800_B8","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1111\/biom.12021","article-title":"Joint high-dimensional Bayesian variable and covariance selection with an application to eQTL analysis","volume":"69","author":"Bhadra,","year":"2013","journal-title":"Biometrics"},{"key":"2020021310531777800_B9","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1093\/biomet\/asw042","article-title":"Fast sampling with Gaussian scale mixture priors in high-dimensional regression","volume":"103","author":"Bhattacharya,","year":"2016","journal-title":"Biometrika"},{"key":"2020021310531777800_B10","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1093\/biomet\/asr013","article-title":"Sparse Bayesian infinite factor models","volume":"98","author":"Bhattacharya,","year":"2011","journal-title":"Biometrika"},{"key":"2020021310531777800_B11","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1214\/18-AOS1712","article-title":"Bayesian fractional posteriors","volume":"47","author":"Bhattacharya,","year":"2019","journal-title":"Ann. Statist."},{"key":"2020021310531777800_B12","doi-asserted-by":"crossref","first-page":"1610","DOI":"10.1080\/01621459.2012.716344","article-title":"Consistent high-dimensional Bayesian variable selection via penalized credible regions","volume":"107","author":"Bondell,","year":"2012","journal-title":"J. Am. Statist. Assoc."},{"key":"2020021310531777800_B13","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1111\/1467-9868.00144","article-title":"Multivariate Bayesian variable selection and prediction","volume":"60","author":"Brown,","year":"1998","journal-title":"J. R. Statist. Soc."},{"key":"2020021310531777800_B14","doi-asserted-by":"crossref","first-page":"1282","DOI":"10.1214\/11-AOS876","article-title":"Optimal selection of reduced rank estimators of high-dimensional matrices","volume":"39","author":"Bunea,","year":"2011","journal-title":"Ann. Statist."},{"key":"2020021310531777800_B15","doi-asserted-by":"crossref","first-page":"2359","DOI":"10.1214\/12-AOS1039","article-title":"Joint variable and rank selection for parsimonious estimation of high-dimensional matrices","volume":"40","author":"Bunea,","year":"2012","journal-title":"Ann. Statist."},{"key":"2020021310531777800_B16","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1093\/biomet\/asq017","article-title":"The horseshoe estimator for sparse signals","volume":"97","author":"Carvalho,","year":"2010","journal-title":"Biometrika"},{"key":"2020021310531777800_B17","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1093\/biomet\/ast036","article-title":"Reduced rank regression via adaptive nuclear norm penalization","volume":"100","author":"Chen,","year":"2013","journal-title":"Biometrika"},{"key":"2020021310531777800_B18","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.1080\/01621459.2012.734178","article-title":"Sparse reduced-rank regression for simultaneous dimension reduction and variable selection","volume":"107","author":"Chen,","year":"2012","journal-title":"J. Am. Statist. Assoc."},{"key":"2020021310531777800_B19","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1111\/j.1467-9868.2009.00723.x","article-title":"Sparse partial least squares regression for simultaneous dimension reduction and variable selection","volume":"72","author":"Chun,","year":"2010","journal-title":"J. R. Statist. Soc."},{"key":"2020021310531777800_B20","doi-asserted-by":"crossref","first-page":"881","DOI":"10.1080\/01621459.1993.10476353","article-title":"Variable selection via Gibbs sampling","volume":"88","author":"George,","year":"1993","journal-title":"J. Am. Statist. Assoc."},{"key":"2020021310531777800_B21","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0304-4076(95)01773-9","article-title":"Bayesian reduced rank regression in econometrics","volume":"75","author":"Geweke,","year":"1996","journal-title":"J. Economet."},{"key":"2020021310531777800_B22","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1214\/aos\/1016218228","article-title":"Convergence rates of posterior distributions","volume":"28","author":"Ghosal,","year":"2000","journal-title":"Ann. Statist."},{"key":"2020021310531777800_B23","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1214\/16-BA1029","article-title":"Asymptotic optimality of one-group shrinkage priors in sparse high-dimensional problems","volume":"12","author":"Ghosh,","year":"2017","journal-title":"Bayesian Anal."},{"key":"2020021310531777800_B24","volume-title":"Matrix Computations","author":"Golub,","year":"1996","edition":"3rd ed"},{"key":"2020021310531777800_B25","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1093\/biomet\/82.4.711","article-title":"Reversible jump Markov chain Monte Carlo computation and Bayesian model determination","volume":"82","author":"Green,","year":"1995","journal-title":"Biometrika"},{"key":"2020021310531777800_B26","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1080\/01621459.2014.993077","article-title":"Decoupling shrinkage and selection in Bayesian linear models: A posterior summary perspective","volume":"110","author":"Hahn,","year":"2015","journal-title":"J. Am. Statist. Assoc."},{"key":"2020021310531777800_B27","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1198\/jcgs.2009.07177","article-title":"Simulation of the matrix Bingham\u2013von Mises-Fisher distribution, with applications to multivariate and relational data","volume":"18","author":"Hoff,","year":"2009","journal-title":"J. Comp. Graph. Statist."},{"key":"2020021310531777800_B28","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/0047-259X(75)90042-1","article-title":"Reduced-rank regression for the multivariate linear model","volume":"5","author":"Izenman,","year":"1975","journal-title":"J. Mult. Anal."},{"key":"2020021310531777800_B29","doi-asserted-by":"crossref","first-page":"1487","DOI":"10.1214\/009053607000000019","article-title":"Bayesian variable selection for high-dimensional generalized linear models: Convergence rates of the fitted densities","volume":"35","author":"Jiang,","year":"2007","journal-title":"Ann. Statist."},{"key":"2020021310531777800_B30","volume-title":"arXiv:","author":"Kundu,","year":"2013"},{"key":"2020021310531777800_B31","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1126\/science.1075090","article-title":"Transcriptional regulatory networks in saccharomyces cerevisiae","volume":"298","author":"Lee,","year":"2002","journal-title":"Science"},{"key":"2020021310531777800_B32","doi-asserted-by":"crossref","first-page":"3396","DOI":"10.1109\/TIT.2006.878172","article-title":"Information theory and mixing least-squares regressions","volume":"52","author":"Leung,","year":"2006","journal-title":"IEEE Trans. Info. Theory"},{"key":"2020021310531777800_B33","volume-title":"Proc. KDD Cup and Workshop","author":"Lim,","year":"2007"},{"key":"2020021310531777800_B34","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1080\/01621459.1999.10473879","article-title":"Parameter expansion for data augmentation","volume":"94","author":"Liu,","year":"1999","journal-title":"J. Am. Statist. Assoc."},{"key":"2020021310531777800_B35","first-page":"41","article-title":"Bayesian model assessment in factor analysis","volume":"14","author":"Lopes,","year":"2004","journal-title":"Statist. Sinica"},{"key":"2020021310531777800_B36","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1017\/CBO9780511584589.009","volume-title":"Bayesian Inference for Gene Expression and Proteomics","author":"Lucas,","year":"2006"},{"key":"2020021310531777800_B37","doi-asserted-by":"crossref","first-page":"1102","DOI":"10.1214\/14-AOS1215","article-title":"Posterior contraction in sparse Bayesian factor models for massive covariance matrices","volume":"42","author":"Pati,","year":"2014","journal-title":"Ann. Statist."},{"key":"2020021310531777800_B38","first-page":"501","article-title":"Shrink globally, act locally: Sparse Bayesian regularization and prediction","volume":"9","author":"Polson,","year":"2010","journal-title":"Bayesian Statist."},{"key":"2020021310531777800_B39","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1111\/rssb.12042","article-title":"The Bayesian bridge","volume":"76","author":"Polson,","year":"2014","journal-title":"J. R. Statist. Soc."},{"key":"2020021310531777800_B40","volume-title":"R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing","year":"2019"},{"key":"2020021310531777800_B41","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1111\/1467-9868.00288","article-title":"Fast sampling of Gaussian Markov random fields","volume":"63","author":"Rue,","year":"2001","journal-title":"J. R. Statist. Soc."},{"key":"2020021310531777800_B42","doi-asserted-by":"crossref","first-page":"618","DOI":"10.1093\/biostatistics\/kxx007","article-title":"Efficient inference for genetic association studies with multiple outcomes","volume":"18","author":"Ruffieux,","year":"2017","journal-title":"Biostatistics"},{"key":"2020021310531777800_B43","volume-title":"Proc. 25th Int. Conf. Machine learning","author":"Salakhutdinov,","year":"2008"},{"key":"2020021310531777800_B44","doi-asserted-by":"crossref","first-page":"2587","DOI":"10.1214\/10-AOS792","article-title":"Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem","volume":"38","author":"Scott,","year":"2010","journal-title":"Ann. Statist."},{"key":"2020021310531777800_B45","doi-asserted-by":"crossref","first-page":"3273","DOI":"10.1091\/mbc.9.12.3273","article-title":"Comprehensive identification of cell cycle\u2013regulated genes of the yeast saccharomyces cerevisiae by microarray hybridization","volume":"9","author":"Spellman,","year":"1998","journal-title":"Molec. Biol. Cell"},{"key":"2020021310531777800_B46","doi-asserted-by":"crossref","first-page":"2585","DOI":"10.1214\/14-EJS962","article-title":"The horseshoe estimator: Posterior concentration around nearly black vectors","volume":"8","author":"van der Pas,","year":"2014","journal-title":"Electron. J. Statist."},{"key":"2020021310531777800_B47","volume-title":"Multivariate Reduced-Rank Regression: Theory and Applications","author":"Velu,","year":"2013"},{"key":"2020021310531777800_B48","doi-asserted-by":"crossref","first-page":"2866","DOI":"10.1016\/j.csda.2010.03.028","article-title":"Sparse seemingly unrelated regression modelling: Applications in finance and econometrics","volume":"54","author":"Wang,","year":"2010","journal-title":"Comp. Statist. Data Anal."},{"key":"2020021310531777800_B49","doi-asserted-by":"crossref","first-page":"1486","DOI":"10.1093\/bioinformatics\/btm125","article-title":"Group SCAD regression analysis for microarray time course gene expression data","volume":"23","author":"Wang,","year":"2007","journal-title":"Bioinformatics"},{"key":"2020021310531777800_B50","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1111\/j.1467-9868.2007.00591.x","article-title":"Dimension reduction and coefficient estimation in multivariate linear regression","volume":"69","author":"Yuan,","year":"2007","journal-title":"J. R. Statist. Soc."},{"key":"2020021310531777800_B51","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1111\/j.1467-9868.2005.00532.x","article-title":"Model selection and estimation in regression with grouped variables","volume":"68","author":"Yuan,","year":"2006","journal-title":"J. R. Statist. Soc."},{"key":"2020021310531777800_B52","doi-asserted-by":"crossref","first-page":"1418","DOI":"10.1198\/016214506000000735","article-title":"The adaptive lasso and its oracle properties","volume":"101","author":"Zou,","year":"2006","journal-title":"J. Am. Statist. Assoc."}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/107\/1\/205\/32450848\/asz056.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/107\/1\/205\/32450848\/asz056.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,2,13]],"date-time":"2020-02-13T10:53:37Z","timestamp":1581591217000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/107\/1\/205\/5638937"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,23]]},"references-count":52,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,11,23]]},"published-print":{"date-parts":[[2020,3,1]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asz056","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2020,3]]},"published":{"date-parts":[[2019,11,23]]}}}