{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T08:31:21Z","timestamp":1773304281236,"version":"3.50.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T00:00:00Z","timestamp":1651017600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T00:00:00Z","timestamp":1651017600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Data Anal Classif"],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Sparse PCA methods are used to overcome the difficulty of interpreting the solution obtained from PCA. However, constraining PCA to obtain sparse solutions is an intractable problem, especially in a high-dimensional setting. Penalized methods are used to obtain sparse solutions due to their computational tractability. Nevertheless, recent developments permit efficiently obtaining good solutions of cardinality-constrained PCA problems allowing comparison between these approaches. Here, we conduct a comparison between a penalized PCA method with its cardinality-constrained counterpart for the least-squares formulation of PCA imposing sparseness on the component weights. We compare the penalized and cardinality-constrained methods through a simulation study that estimates the sparse structure\u2019s recovery, mean absolute bias, mean variance, and mean squared error. Additionally, we use a high-dimensional data set to illustrate the methods in practice. Results suggest that using cardinality-constrained methods leads to better recovery of the sparse structure.<\/jats:p>","DOI":"10.1007\/s11634-022-00499-2","type":"journal-article","created":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T08:03:50Z","timestamp":1651046630000},"page":"269-286","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Sparsifying the least-squares approach to PCA: comparison of lasso and cardinality constraint"],"prefix":"10.1007","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2709-0363","authenticated-orcid":false,"given":"Rosember","family":"Guerra-Urzola","sequence":"first","affiliation":[]},{"given":"Niek C.","family":"de Schipper","sequence":"additional","affiliation":[]},{"given":"Anya","family":"Tonne","sequence":"additional","affiliation":[]},{"given":"Klaas","family":"Sijtsma","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1394-3422","authenticated-orcid":false,"given":"Juan C.","family":"Vera","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2271-793X","authenticated-orcid":false,"given":"Katrijn","family":"Van Deun","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,27]]},"reference":[{"key":"499_CR1","unstructured":"Adachi K, Kiers HAL (2017) Sparse regression without using a penalty function. http:\/\/www.jfssa.jp\/taikai\/2017\/table\/program_detail\/pdf\/1-50\/10009.pdf"},{"issue":"4","key":"499_CR2","doi-asserted-by":"publisher","first-page":"1403","DOI":"10.1007\/s00180-015-0608-4","volume":"31","author":"K Adachi","year":"2016","unstructured":"Adachi K, Trendafilov NT (2016) Sparse principal component analysis subject to prespecified cardinality of loadings. Comput Stat 31(4):1403\u20131427","journal-title":"Comput Stat"},{"issue":"3","key":"499_CR3","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s12532-018-0153-6","volume":"11","author":"L Berk","year":"2019","unstructured":"Berk L, Bertsimas D (2019) Certifiably optimal sparse principal component analysis. Math Program Comput 11(3):381\u2013420","journal-title":"Math Program Comput"},{"issue":"1","key":"499_CR4","doi-asserted-by":"publisher","first-page":"300","DOI":"10.1214\/18-AOS1804","volume":"48","author":"D Bertsimas","year":"2020","unstructured":"Bertsimas D, Parys BV (2020) Sparse high-dimensional regression: exact scalable algorithms and phase transitions. Ann Stat 48(1):300\u2013323","journal-title":"Ann Stat"},{"issue":"2","key":"499_CR5","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1214\/15-AOS1388","volume":"44","author":"D Bertsimas","year":"2016","unstructured":"Bertsimas D, King A, Mazumder R (2016) Best subset selection via a modern optimization lens. Ann Stat 44(2):813\u2013852","journal-title":"Ann Stat"},{"key":"499_CR6","doi-asserted-by":"publisher","first-page":"103907","DOI":"10.1016\/j.chemolab.2019.103907","volume":"196","author":"J Camacho","year":"2020","unstructured":"Camacho J, Smilde A, Saccenti E, Westerhuis J (2020) All sparse pca models are wrong, but some are useful. Part i: computation of scores, residuals and explained variance. Chemomet Intell Lab Syst 196:103907","journal-title":"Chemomet Intell Lab Syst"},{"key":"499_CR7","doi-asserted-by":"publisher","first-page":"104212","DOI":"10.1016\/j.chemolab.2020.104212","volume":"208","author":"J Camacho","year":"2021","unstructured":"Camacho J, Smilde A, Saccenti E, Westerhuis J, Bro R (2021) All sparse pca models are wrong, but some are useful. Part ii: limitations and problems of deflation. Chemomet Intell Lab Syst 208:104212","journal-title":"Chemomet Intell Lab Syst"},{"key":"499_CR8","doi-asserted-by":"crossref","unstructured":"d\u2019Aspremont A, El Ghaoui L, Jordan MI, Lanckriet GRG (2007) A direct formulation for sparse pca using semidefinite programming. SIAM Rev 49(3):434\u2013448","DOI":"10.1137\/050645506"},{"key":"499_CR9","unstructured":"d\u2019Aspremont A, Bach F, El Ghaoui L (2008) Optimal solutions for sparse principal component analysis. J Mach Learn Res 9(7):1269\u20131294"},{"issue":"2","key":"499_CR10","doi-asserted-by":"publisher","first-page":"977","DOI":"10.1137\/18M1211350","volume":"80","author":"NB Erichson","year":"2020","unstructured":"Erichson NB, Zheng P, Manohar K, Brunton SL, Kutz JN, Aravkin AY (2020) Sparse principal component analysis via variable projection. SIAM J Appl Math 80(2):977\u20131002","journal-title":"SIAM J Appl Math"},{"key":"499_CR11","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1016\/j.chemolab.2016.07.013","volume":"158","author":"Z Gu","year":"2016","unstructured":"Gu Z, Van Deun K (2016) A variable selection method for simultaneous component based data integration. Chemom Intell Lab Syst 158:187\u2013199","journal-title":"Chemom Intell Lab Syst"},{"issue":"1","key":"499_CR12","doi-asserted-by":"publisher","first-page":"18608","DOI":"10.1038\/s41598-019-54673-2","volume":"9","author":"Z Gu","year":"2019","unstructured":"Gu Z, de Schipper NC, Deun KV (2019) Variable selection in the regularized simultaneous component analysis method for multi-source data integration. Sci Rep 9(1):18608","journal-title":"Sci Rep"},{"issue":"4","key":"499_CR13","doi-asserted-by":"publisher","first-page":"893","DOI":"10.1007\/s11336-021-09773-2","volume":"86","author":"R Guerra-Urzola","year":"2021","unstructured":"Guerra-Urzola R, Van Deun K, Lizcano JV, Sijtsma K (2021) A guide for sparse pca: model comparison and applications. Psychometrika 86(4):893\u2013919","journal-title":"Psychometrika"},{"key":"499_CR14","unstructured":"Hastie T, Tibshirani R, Tibshirani RJ (2017) Extended comparisons of best subset selection, forward stepwise selection, and the lasso. arXiv:1707.08692"},{"issue":"19","key":"499_CR15","doi-asserted-by":"publisher","first-page":"5052","DOI":"10.1109\/TSP.2016.2576427","volume":"64","author":"K Huang","year":"2016","unstructured":"Huang K, Sidiropoulos ND, Liavas AP (2016) A flexible and efficient algorithmic framework for constrained matrix and tensor factorization. IEEE Trans Signal Process 64(19):5052\u20135065","journal-title":"IEEE Trans Signal Process"},{"issue":"1","key":"499_CR16","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1198\/0003130042836","volume":"58","author":"DR Hunter","year":"2004","unstructured":"Hunter DR, Lange K (2004) A tutorial on mm algorithms. Am Stat 58(1):30\u201337","journal-title":"Am Stat"},{"key":"499_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1904-8","volume-title":"Principal component analysis","author":"IT Jolliffe","year":"1986","unstructured":"Jolliffe IT (1986) Principal component analysis. Springer, New York"},{"key":"499_CR18","doi-asserted-by":"publisher","unstructured":"Jolliffe IT (2002) Principal components in regression analysis. In: Principal component analysis. Springer Series in Statistics, Springer, New York, NY. https:\/\/doi.org\/10.1007\/0-387-22440-8_8","DOI":"10.1007\/0-387-22440-8_8"},{"issue":"2","key":"499_CR19","first-page":"517","volume":"11","author":"M Journ\u00e9e","year":"2010","unstructured":"Journ\u00e9e M, Nesterov Y, Richt\u00e1rik P, Sepulchre R (2010) Generalized power method for sparse principal component analysis. J Mach Learn Res 11(2):517\u2013553","journal-title":"J Mach Learn Res"},{"issue":"1","key":"499_CR20","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/S0167-9473(02)00142-1","volume":"41","author":"HA Kiers","year":"2002","unstructured":"Kiers HA (2002) Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems. Comput Stat Data Anal 41(1):157\u2013170","journal-title":"Comput Stat Data Anal"},{"issue":"2","key":"499_CR21","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1137\/S0097539792240406","volume":"24","author":"BK Natarajan","year":"1995","unstructured":"Natarajan BK (1995) Sparse approximate solutions to linear systems. SIAM J Comput 24(2):227\u2013234","journal-title":"SIAM J Comput"},{"issue":"14","key":"499_CR22","doi-asserted-by":"publisher","first-page":"1682","DOI":"10.1093\/hmg\/ddm116","volume":"16","author":"Y Nishimura","year":"2007","unstructured":"Nishimura Y, Martin CL, Vazquez-Lopez A, Spence SJ, Alvarez-Retuerto AI, Sigman M, Steindler C, Pellegrini S, Schanen NC, Warren ST et al (2007) Genome-wide expression profiling of lymphoblastoid cell lines distinguishes different forms of autism and reveals shared pathways. Hum Mol Genet 16(14):1682\u20131698","journal-title":"Hum Mol Genet"},{"key":"499_CR23","unstructured":"R Core Team (2020) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria"},{"issue":"3","key":"499_CR24","doi-asserted-by":"publisher","first-page":"1493","DOI":"10.1007\/s11081-020-09562-3","volume":"22","author":"P Richt\u00e1rik","year":"2021","unstructured":"Richt\u00e1rik P, Jahani M, Ahipa\u015fao\u011flu SD, Tak\u00e1\u010d M (2021) Alternating maximization: unifying framework for 8 sparse pca formulations and efficient parallel codes. Optim Eng 22(3):1493\u20131519","journal-title":"Optim Eng"},{"issue":"6","key":"499_CR25","doi-asserted-by":"publisher","first-page":"1015","DOI":"10.1016\/j.jmva.2007.06.007","volume":"99","author":"H Shen","year":"2008","unstructured":"Shen H, Huang JZ (2008) Sparse principal component analysis via regularized low rank matrix approximation. J Multivar Anal 99(6):1015\u20131034","journal-title":"J Multivar Anal"},{"key":"499_CR26","volume-title":"Least squares optimization in multivariate analysis","author":"JM ten Berge","year":"1993","unstructured":"ten Berge JM (1993) Least squares optimization in multivariate analysis. DSWO Press, Leiden University Leiden, Leiden"},{"issue":"1","key":"499_CR27","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani R (1996) Regression shrinkage and selection via the lasso. J Roy Stat Soc Ser B (Methodol) 58(1):267\u2013288","journal-title":"J Roy Stat Soc Ser B (Methodol)"},{"issue":"3","key":"499_CR28","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1111\/j.1467-9868.2011.00771.x","volume":"73","author":"R Tibshirani","year":"2011","unstructured":"Tibshirani R (2011) Regression shrinkage and selection via the lasso: a retrospective. J Roy Stat Soc Ser B (Methodol) 73(3):273\u2013282","journal-title":"J Roy Stat Soc Ser B (Methodol)"},{"issue":"3\u20134","key":"499_CR29","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1007\/s00180-013-0434-5","volume":"29","author":"NT Trendafilov","year":"2014","unstructured":"Trendafilov NT (2014) From simple structure to sparse components: a review. Comput Stat 29(3\u20134):431\u2013454","journal-title":"Comput Stat"},{"issue":"3","key":"499_CR30","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1023\/A:1017501703105","volume":"109","author":"P Tseng","year":"2001","unstructured":"Tseng P (2001) Convergence of a block coordinate descent method for nondifferentiable minimization. J Optim Theory Appl 109(3):475\u2013494","journal-title":"J Optim Theory Appl"},{"issue":"1","key":"499_CR31","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1186\/1471-2105-10-246","volume":"10","author":"K Van Deun","year":"2009","unstructured":"Van Deun K, Smilde AK, van der Werf MJ, Kiers HA, Van Mechelen I (2009) A structured overview of simultaneous component based data integration. BMC Bioinf 10(1):246","journal-title":"BMC Bioinf"},{"issue":"4","key":"499_CR32","doi-asserted-by":"publisher","first-page":"923","DOI":"10.1080\/10618600.2013.858632","volume":"23","author":"D Yang","year":"2014","unstructured":"Yang D, Ma Z, Buja A (2014) A sparse singular value decomposition method for high-dimensional data. J Comput Graph Stat 23(4):923\u2013942","journal-title":"J Comput Graph Stat"},{"issue":"2","key":"499_CR33","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1111\/j.1467-9868.2005.00503.x","volume":"67","author":"H Zou","year":"2005","unstructured":"Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J Roy Stat Soc Ser B (Stat Methodol) 67(2):301\u2013320","journal-title":"J Roy Stat Soc Ser B (Stat Methodol)"},{"key":"499_CR34","unstructured":"Zou H, Hastie T (2018) elasticnet: elastic-net for sparse estimation and sparse PCA. R package version 1(1):1"},{"issue":"2","key":"499_CR35","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1198\/106186006X113430","volume":"15","author":"H Zou","year":"2006","unstructured":"Zou H, Hastie T, Tibshirani R (2006) Sparse principal component analysis. J Comput Graph Stat 15(2):265\u2013286","journal-title":"J Comput Graph Stat"}],"container-title":["Advances in Data Analysis and Classification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11634-022-00499-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11634-022-00499-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11634-022-00499-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,23]],"date-time":"2024-09-23T01:50:39Z","timestamp":1727056239000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11634-022-00499-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,27]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["499"],"URL":"https:\/\/doi.org\/10.1007\/s11634-022-00499-2","relation":{},"ISSN":["1862-5347","1862-5355"],"issn-type":[{"value":"1862-5347","type":"print"},{"value":"1862-5355","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,27]]},"assertion":[{"value":"9 March 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 January 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 March 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 April 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}