{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T19:01:43Z","timestamp":1770836503278,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T00:00:00Z","timestamp":1632096000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T00:00:00Z","timestamp":1632096000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["678304"],"award-info":[{"award-number":["678304"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["666992"],"award-info":[{"award-number":["666992"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["826421"],"award-info":[{"award-number":["826421"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-19-P3IA-0001"],"award-info":[{"award-number":["ANR-19-P3IA-0001"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-10-IAIHU-06"],"award-info":[{"award-number":["ANR-10-IAIHU-06"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["SN COMPUT. SCI."],"published-print":{"date-parts":[[2021,11]]},"DOI":"10.1007\/s42979-021-00865-5","type":"journal-article","created":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T19:15:46Z","timestamp":1632165346000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Mixture of Conditional Gaussian Graphical Models for Unlabelled Heterogeneous Populations in the Presence of Co-factors"],"prefix":"10.1007","volume":"2","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7820-0032","authenticated-orcid":false,"given":"Thomas","family":"Lartigue","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9450-6920","authenticated-orcid":false,"given":"Stanley","family":"Durrleman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5692-4945","authenticated-orcid":false,"given":"St\u00e9phanie","family":"Allassonni\u00e8re","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,9,20]]},"reference":[{"issue":"6","key":"865_CR1","doi-asserted-by":"publisher","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","volume":"19","author":"H Akaike","year":"1974","unstructured":"Akaike H. A new look at the statistical model identification. IEEE Trans Autom Control. 1974;19(6):716\u201323.","journal-title":"IEEE Trans Autom Control"},{"key":"865_CR2","doi-asserted-by":"crossref","unstructured":"Banerjee O, Ghaoui LE, d\u2019Aspremont A, Natsoulis G. Convex optimization techniques for fitting sparse gaussian graphical models. In: Proceedings of the 23rd international conference on Machine learning, ACM, 2006;89\u201396.","DOI":"10.1145\/1143844.1143856"},{"key":"865_CR3","first-page":"485","volume":"9","author":"O Banerjee","year":"2008","unstructured":"Banerjee O, Ghaoui LE, d\u2019Aspremont A. Model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data. J Mach Learning Res. 2008;9:485\u2013516.","journal-title":"J Mach Learning Res"},{"key":"865_CR4","doi-asserted-by":"publisher","first-page":"294","DOI":"10.3389\/fgene.2013.00294","volume":"4","author":"H Chun","year":"2013","unstructured":"Chun H, Chen M, Li B, Zhao H. Joint conditional gaussian graphical models with multiple sources of genomic data. Front Genet. 2013;4:294.","journal-title":"Front Genet"},{"key":"865_CR5","doi-asserted-by":"crossref","unstructured":"Combettes PL, Pesquet JC. Proximal splitting methods in signal processing. In: Fixed-point algorithms for inverse problems in science and engineering, Springer, 2011; 185\u2013212.","DOI":"10.1007\/978-1-4419-9569-8_10"},{"issue":"2","key":"865_CR6","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1111\/rssb.12033","volume":"76","author":"P Danaher","year":"2014","unstructured":"Danaher P, Wang P, Witten DM. The joint graphical lasso for inverse covariance estimation across multiple classes. J Royal Stat Soc Series B. 2014;76(2):373\u201397.","journal-title":"J Royal Stat Soc Series B"},{"key":"865_CR7","doi-asserted-by":"crossref","unstructured":"Dempster AP. Covariance selection. Biometrics 1972; 157\u2013175.","DOI":"10.2307\/2528966"},{"issue":"2","key":"865_CR8","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/BF01897167","volume":"5","author":"WS DeSarbo","year":"1988","unstructured":"DeSarbo WS, Cron WL. A maximum likelihood methodology for clusterwise linear regression. J Classif. 1988;5(2):249\u201382.","journal-title":"J Classif"},{"issue":"3","key":"865_CR9","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1109\/34.990138","volume":"24","author":"MAT Figueiredo","year":"2002","unstructured":"Figueiredo MAT, Jain AK. Unsupervised learning of finite mixture models. IEEE Trans Pattern Anal Mach Intell. 2002;24(3):381\u201396.","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"issue":"4","key":"865_CR10","doi-asserted-by":"publisher","first-page":"791","DOI":"10.1007\/s11222-018-9838-y","volume":"29","author":"M Fop","year":"2019","unstructured":"Fop M, Murphy TB, Scrucca L. Model-based clustering with sparse covariance matrices. Stat Comput. 2019;29(4):791\u2013819.","journal-title":"Stat Comput"},{"issue":"3","key":"865_CR11","doi-asserted-by":"publisher","first-page":"432","DOI":"10.1093\/biostatistics\/kxm045","volume":"9","author":"J Friedman","year":"2008","unstructured":"Friedman J, Hastie T, Tibshirani R. Sparse inverse covariance estimation with the graphical lasso. Biostatistics. 2008;9(3):432\u201341.","journal-title":"Biostatistics"},{"key":"865_CR12","first-page":"1133","volume":"10","author":"C Gao","year":"2016","unstructured":"Gao C, Zhu Y, Shen X, Pan W. Estimation of multiple networks in gaussian mixture models. Electron J Stat. 2016;10:1133.","journal-title":"Electron J Stat"},{"issue":"1","key":"865_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1093\/biomet\/asq060","volume":"98","author":"J Guo","year":"2011","unstructured":"Guo J, Levina E, Michailidis G, Zhu J. Joint estimation of multiple graphical models. Biometrika. 2011;98(1):1\u201315.","journal-title":"Biometrika"},{"issue":"1","key":"865_CR14","first-page":"7981","volume":"18","author":"B Hao","year":"2017","unstructured":"Hao B, Sun WW, Liu Y, Cheng G. Simultaneous clustering and estimation of heterogeneous graphical models. J Mach Learning Res. 2017;18(1):7981\u20138038.","journal-title":"J Mach Learning Res"},{"key":"865_CR15","doi-asserted-by":"publisher","first-page":"622","DOI":"10.1016\/j.trc.2017.12.007","volume":"86","author":"Y Hara","year":"2018","unstructured":"Hara Y, Suzuki J, Kuwahara M. Network-wide traffic state estimation using a mixture gaussian graphical model and graphical lasso. Transport Res Part C Emerging Technol. 2018;86:622\u201338.","journal-title":"Transport Res Part C Emerging Technol"},{"key":"865_CR16","unstructured":"Honorio J, Samaras D. Multi-task learning of gaussian graphical models. In: ICML, Citeseer, 2010; 447\u2013454."},{"issue":"7","key":"865_CR17","first-page":"3034","volume":"29","author":"F Huang","year":"2018","unstructured":"Huang F, Chen S, Huang SJ. Joint estimation of multiple conditional Gaussian graphical models. IEEE Trans Neural Netw Learning Syst. 2018;29(7):3034\u201346.","journal-title":"IEEE Trans Neural Netw Learning Syst"},{"issue":"2","key":"865_CR18","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1093\/biomet\/76.2.297","volume":"76","author":"CM Hurvich","year":"1989","unstructured":"Hurvich CM, Tsai CL. Regression and time series model selection in small samples. Biometrika. 1989;76(2):297\u2013307.","journal-title":"Biometrika"},{"issue":"2","key":"865_CR19","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1162\/neco.1994.6.2.181","volume":"6","author":"MI Jordan","year":"1994","unstructured":"Jordan MI, Jacobs RA. Hierarchical mixtures of experts and the EM algorithm. Neural Comput. 1994;6(2):181\u2013214.","journal-title":"Neural Comput"},{"issue":"479","key":"865_CR20","doi-asserted-by":"publisher","first-page":"1025","DOI":"10.1198\/016214507000000590","volume":"102","author":"A Khalili","year":"2007","unstructured":"Khalili A, Chen J. Variable selection in finite mixture of regression models. J Am Stat Assoc. 2007;102(479):1025\u201338.","journal-title":"J Am Stat Assoc"},{"issue":"1","key":"865_CR21","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s10489-015-0691-9","volume":"44","author":"M Kim","year":"2016","unstructured":"Kim M. Sparse inverse covariance learning of conditional Gaussian mixtures for multiple-output regression. Appl Intell. 2016;44(1):17\u201329.","journal-title":"Appl Intell"},{"key":"865_CR22","unstructured":"Krishnamurthy A. High-dimensional clustering with sparse gaussian mixture models. Unpublished paper 2011; 191\u2013192."},{"issue":"1","key":"865_CR23","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1214\/aoms\/1177729694","volume":"22","author":"S Kullback","year":"1951","unstructured":"Kullback S, Leibler RA. On information and sufficiency. Ann Math Stat. 1951;22(1):79\u201386.","journal-title":"Ann Math Stat"},{"key":"865_CR24","unstructured":"Lartigue T. Mixture of gaussian graphical models with constraints. PhD thesis, Institut Polytechnique de Paris 2020."},{"issue":"1","key":"865_CR25","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1080\/00949650701611143","volume":"79","author":"H Lee","year":"2009","unstructured":"Lee H, Ghosh SK. Performance of information criteria for spatial models. J Stat Comput Simul. 2009;79(1):93\u2013106.","journal-title":"J Stat Comput Simul"},{"issue":"1","key":"865_CR26","first-page":"445","volume":"15","author":"K Mohan","year":"2014","unstructured":"Mohan K, London P, Fazel M, Witten D, Lee SI. Node-based learning of multiple Gaussian graphical models. J Mach Learning Res. 2014;15(1):445\u201388.","journal-title":"J Mach Learning Res"},{"key":"865_CR27","doi-asserted-by":"crossref","unstructured":"Ou-Yang L, Zhang XF, Hu X, Yan H. Differential network analysis via weighted fused conditional gaussian graphical model. IEEE\/ACM transactions on computational biology and bioinformatics 2019.","DOI":"10.1109\/TCBB.2019.2924418"},{"key":"865_CR28","doi-asserted-by":"crossref","unstructured":"Schiratti JB, Allassonniere S, Routier A, Colliot O, Durrleman S, Initiative ADN, et\u00a0al. A mixed-effects model with time reparametrization for longitudinal univariate manifold-valued data. In: International Conference on Information Processing in Medical Imaging, Springer, 2015; 564\u2013575.","DOI":"10.1007\/978-3-319-19992-4_44"},{"issue":"2","key":"865_CR29","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1214\/aos\/1176344136","volume":"6","author":"G Schwarz","year":"1978","unstructured":"Schwarz G, et al. Estimating the dimension of a model. Ann Stat. 1978;6(2):461\u20134.","journal-title":"Ann Stat"},{"key":"865_CR30","unstructured":"Sohn KA, Kim S. Joint estimation of structured sparsity and output structure in multiple-output regression via inverse-covariance regularization. In: Artificial Intelligence and Statistics, 2012; 1081\u20131089."},{"key":"865_CR31","unstructured":"Varoquaux G, Gramfort A, Poline JB, Thirion B. Brain covariance selection: better individual functional connectivity models using population prior. In: Advances in neural information processing systems, 2010; 2334\u20132342."},{"key":"865_CR32","unstructured":"Wytock M, Kolter Z. Sparse gaussian conditional random fields: algorithms, theory, and application to energy forecasting. In: International conference on machine learning, 2013; 1265\u20131273."},{"issue":"11","key":"865_CR33","doi-asserted-by":"publisher","first-page":"3950","DOI":"10.1016\/j.patcog.2012.04.031","volume":"45","author":"MS Yang","year":"2012","unstructured":"Yang MS, Lai CY, Lin CY. A robust EM clustering algorithm for Gaussian mixture models. Pattern Recogn. 2012;45(11):3950\u201361.","journal-title":"Pattern Recogn"},{"issue":"2","key":"865_CR34","doi-asserted-by":"publisher","first-page":"916","DOI":"10.1137\/130936397","volume":"25","author":"S Yang","year":"2015","unstructured":"Yang S, Lu Z, Shen X, Wonka P, Ye J. Fused multiple graphical lasso. SIAM J Optim. 2015;25(2):916\u201343.","journal-title":"SIAM J Optim"},{"issue":"4","key":"865_CR35","doi-asserted-by":"publisher","first-page":"2630","DOI":"10.1214\/11-AOAS494","volume":"5","author":"J Yin","year":"2011","unstructured":"Yin J, Li H. A sparse conditional gaussian graphical model for analysis of genetical genomics data. Ann Appl Stat. 2011;5(4):2630.","journal-title":"Ann Appl Stat"},{"issue":"1","key":"865_CR36","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1111\/j.1467-9868.2005.00532.x","volume":"68","author":"M Yuan","year":"2006","unstructured":"Yuan M, Lin Y. Model selection and estimation in regression with grouped variables. J Royal Stat Soc: Series B. 2006;68(1):49\u201367.","journal-title":"J Royal Stat Soc: Series B"},{"issue":"1","key":"865_CR37","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1093\/biomet\/asm018","volume":"94","author":"M Yuan","year":"2007","unstructured":"Yuan M, Lin Y. Model selection and estimation in the Gaussian graphical model. Biometrika. 2007;94(1):19\u201335.","journal-title":"Biometrika"},{"key":"865_CR38","doi-asserted-by":"publisher","first-page":"1473","DOI":"10.1214\/09-EJS487","volume":"3","author":"H Zhou","year":"2009","unstructured":"Zhou H, Pan W, Shen X. Penalized model-based clustering with unconstrained covariance matrices. Electron J Stat. 2009;3:1473.","journal-title":"Electron J Stat"}],"container-title":["SN Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42979-021-00865-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s42979-021-00865-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42979-021-00865-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,30]],"date-time":"2021-10-30T20:18:22Z","timestamp":1635625102000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s42979-021-00865-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,20]]},"references-count":38,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,11]]}},"alternative-id":["865"],"URL":"https:\/\/doi.org\/10.1007\/s42979-021-00865-5","relation":{},"ISSN":["2662-995X","2661-8907"],"issn-type":[{"value":"2662-995X","type":"print"},{"value":"2661-8907","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,20]]},"assertion":[{"value":"5 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 September 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 September 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author states that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Code for our algorithm, as well as a toy example that reproduces some of the results of this paper, publicly available at:","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Code availability"}}],"article-number":"466"}}