{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T10:08:46Z","timestamp":1776247726512,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,2,11]],"date-time":"2022-02-11T00:00:00Z","timestamp":1644537600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,2,11]],"date-time":"2022-02-11T00:00:00Z","timestamp":1644537600000},"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":["Data Min Knowl Disc"],"published-print":{"date-parts":[[2022,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Spectral-based subspace clustering methods have proved successful in many challenging applications such as gene sequencing, image recognition, and motion segmentation. In this work, we first propose a novel spectral-based subspace clustering algorithm that seeks to represent each point as a sparse convex combination of a few nearby points. We then extend the algorithm to a constrained clustering and active learning framework. Our motivation for developing such a framework stems from the fact that typically either a small amount of labelled data are available in advance; or it is possible to label some points at a cost. The latter scenario is typically encountered in the process of validating a cluster assignment. Extensive experiments on simulated and real datasets show that the proposed approach is effective and competitive with state-of-the-art methods.<\/jats:p>","DOI":"10.1007\/s10618-022-00820-9","type":"journal-article","created":{"date-parts":[[2022,2,11]],"date-time":"2022-02-11T07:02:42Z","timestamp":1644562962000},"page":"958-986","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Weighted sparse simplex representation: a unified framework for subspace clustering, constrained clustering, and active learning"],"prefix":"10.1007","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1623-9852","authenticated-orcid":false,"given":"Hankui","family":"Peng","sequence":"first","affiliation":[]},{"given":"Nicos G.","family":"Pavlidis","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,2,11]]},"reference":[{"key":"820_CR1","doi-asserted-by":"crossref","unstructured":"Basu S, Davidson I, Wagstaff K (2008) Constrained clustering: advances in algorithms, theory, and applications. CRC Press","DOI":"10.1201\/9781584889977"},{"key":"820_CR2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex optimization","author":"S Boyd","year":"2004","unstructured":"Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge"},{"issue":"1","key":"820_CR3","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1023\/A:1008324625522","volume":"16","author":"PS Bradley","year":"2000","unstructured":"Bradley PS, Mangasarian OL (2000) $$k$$-plane clustering. J Global Optim 16(1):23\u201332","journal-title":"J Global Optim"},{"issue":"3","key":"820_CR4","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1093\/biomet\/72.3.627","volume":"72","author":"F Critchley","year":"1985","unstructured":"Critchley F (1985) Influence in principal components analysis. Biometrika 72(3):627\u2013636","journal-title":"Biometrika"},{"issue":"11","key":"820_CR5","doi-asserted-by":"publisher","first-page":"2765","DOI":"10.1109\/TPAMI.2013.57","volume":"35","author":"E Elhamifar","year":"2013","unstructured":"Elhamifar E, Vidal R (2013) Sparse subspace clustering: Algorithm, theory, and applications. IEEE Trans Pattern Anal Mach Intell 35(11):2765\u20132781","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"issue":"4","key":"820_CR6","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1080\/10618600.2018.1473777","volume":"27","author":"BR Gaines","year":"2018","unstructured":"Gaines BR, Kim J, Zhou H (2018) Algorithms for fitting the constrained lasso. J Comput Graph Stat 27(4):861\u2013871","journal-title":"J Comput Graph Stat"},{"key":"820_CR7","doi-asserted-by":"crossref","unstructured":"Hu H, Lin Z, Feng J, Zhou J (2014) Smooth representation clustering. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 3834\u20133841","DOI":"10.1109\/CVPR.2014.484"},{"key":"820_CR8","doi-asserted-by":"crossref","unstructured":"Huang H, Yan J, Nie F, Huang J, Cai W, Saykin AJ, Shen L (2013) A new sparse simplex model for brain anatomical and genetic network analysis. In: International conference on medical image computing and computer-assisted intervention. Springer, pp 625\u2013632","DOI":"10.1007\/978-3-642-40763-5_77"},{"key":"820_CR9","unstructured":"Huang J, Nie F, Huang H (2015) A new simplex sparse learning model to measure data similarity for clustering. In: 24th international joint conference on artificial intelligence"},{"issue":"5","key":"820_CR10","doi-asserted-by":"publisher","first-page":"550","DOI":"10.1109\/34.291440","volume":"16","author":"JJ Hull","year":"1994","unstructured":"Hull JJ (1994) A database for handwritten text recognition research. IEEE Trans Pattern Anal Mach Intell 16(5):550\u2013554","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"issue":"1","key":"820_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/1497577.1497578","volume":"3","author":"HP Kriegel","year":"2009","unstructured":"Kriegel HP, Kr\u00f6ger P, Zimek A (2009) Clustering high-dimensional data: A survey on subspace clustering, pattern-based clustering, and correlation clustering. ACM Trans Knowl Discov Data 3(1):1\u201358","journal-title":"ACM Trans Knowl Discov Data"},{"issue":"1\u20132","key":"820_CR12","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1002\/nav.3800020109","volume":"2","author":"HW Kuhn","year":"1955","unstructured":"Kuhn HW (1955) The hungarian method for the assignment problem. Naval Research Logistics Quarterly 2(1\u20132):83\u201397","journal-title":"Naval Research Logistics Quarterly"},{"issue":"11","key":"820_CR13","doi-asserted-by":"publisher","first-page":"2278","DOI":"10.1109\/5.726791","volume":"86","author":"Y LeCun","year":"1998","unstructured":"LeCun Y, Bottou L, Bengio Y, Haffner P (1998) Gradient-based learning applied to document recognition. Proc IEEE 86(11):2278\u20132324","journal-title":"Proc IEEE"},{"issue":"6","key":"820_CR14","doi-asserted-by":"publisher","first-page":"2988","DOI":"10.1109\/TIP.2017.2691557","volume":"26","author":"C Li","year":"2017","unstructured":"Li C, You C, Vidal R (2017) Structured sparse subspace clustering: a joint affinity learning and subspace clustering framework. IEEE Trans Image Process 26(6):2988\u20133001","journal-title":"IEEE Trans Image Process"},{"issue":"6","key":"820_CR15","doi-asserted-by":"publisher","first-page":"1520","DOI":"10.1109\/JSTSP.2018.2867446","volume":"12","author":"C Li","year":"2018","unstructured":"Li C, You C, Vidal R (2018a) On geometric analysis of affine sparse subspace clustering. IEEE J Selected Topics Sig Process 12(6):1520\u20131533","journal-title":"IEEE J Selected Topics Sig Process"},{"key":"820_CR16","doi-asserted-by":"crossref","unstructured":"Li C, Zhang J, Guo J (2018b) Constrained sparse subspace clustering with side-information. In: 2018 24th international conference on pattern recognition. IEEE, pp 2093\u20132099","DOI":"10.1109\/ICPR.2018.8545800"},{"key":"820_CR17","doi-asserted-by":"crossref","unstructured":"Lipor J, Balzano L (2015) Margin-based active subspace clustering. In: 2015 IEEE 6th international workshop on computational advances in multi-sensor adaptive processing. IEEE, pp 377\u2013380","DOI":"10.1109\/CAMSAP.2015.7383815"},{"key":"820_CR18","unstructured":"Lipor J, Balzano L (2017) Leveraging union of subspace structure to improve constrained clustering. In: Proceedings of the 34th international conference on machine learning, JMLR, vol\u00a070, pp 2130\u20132139"},{"issue":"1","key":"820_CR19","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1109\/TPAMI.2012.88","volume":"35","author":"G Liu","year":"2012","unstructured":"Liu G, Lin Z, Yan S, Sun J, Yu Y, Ma Y (2012) Robust recovery of subspace structures by low-rank representation. IEEE Trans Pattern Anal Mach Intell 35(1):171\u2013184","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"issue":"10","key":"820_CR20","doi-asserted-by":"publisher","first-page":"2469","DOI":"10.1109\/TPAMI.2017.2763945","volume":"40","author":"H Liu","year":"2018","unstructured":"Liu H, Tao Z, Fu Y (2018) Partition level constrained clustering. IEEE Trans Pattern Anal Mach Intell 40(10):2469\u20132483","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"820_CR21","doi-asserted-by":"crossref","unstructured":"Lu C, Min H, Zhao Z, Zhu L, Huang D, Yan S (2012) Robust and efficient subspace segmentation via least squares regression. In: European conference on computer vision. Springer, pp 347\u2013360","DOI":"10.1007\/978-3-642-33786-4_26"},{"issue":"3","key":"820_CR22","doi-asserted-by":"publisher","first-page":"736","DOI":"10.1007\/s10618-013-0317-y","volume":"28","author":"B McWilliams","year":"2014","unstructured":"McWilliams B, Montana G (2014) Subspace clustering of high-dimensional data: A predictive approach. Data Min Knowl Disc 28(3):736\u2013772","journal-title":"Data Min Knowl Disc"},{"key":"820_CR23","unstructured":"Ng AY, Jordan MI, Weiss Y (2002) On spectral clustering: Analysis and an algorithm. In: Advances in neural information processing systems, pp 849\u2013856"},{"issue":"3","key":"820_CR24","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1561\/2400000003","volume":"1","author":"N Parikh","year":"2014","unstructured":"Parikh N, Boyd S (2014) Proximal algorithms. Found Trends Optim 1(3):127\u2013239","journal-title":"Found Trends Optim"},{"key":"820_CR25","doi-asserted-by":"crossref","unstructured":"Pelleg D, Baras D (2007) $$K$$-means with large and noisy constraint sets. In: European conference on machine learning. Springer, pp 674\u2013682","DOI":"10.1007\/978-3-540-74958-5_67"},{"key":"820_CR26","doi-asserted-by":"crossref","unstructured":"Peng H, Pavlidis NG (2019) Subspace clustering with active learning. In: IEEE international conference on big data (big data). IEEE, pp 135\u2013144","DOI":"10.1109\/BigData47090.2019.9006361"},{"key":"820_CR27","doi-asserted-by":"crossref","unstructured":"Peng H, Pavlidis NG, Eckley IA, Tsalamanis I (2018) Subspace clustering of very sparse high-dimensional data. In: IEEE international conference on big data (big data). IEEE, pp 3780\u20133783","DOI":"10.1109\/BigData.2018.8622472"},{"issue":"10","key":"820_CR28","doi-asserted-by":"publisher","first-page":"1832","DOI":"10.1109\/TPAMI.2009.191","volume":"32","author":"S Rao","year":"2010","unstructured":"Rao S, Tron R, Vidal R, Ma Y (2010) Motion segmentation in the presence of outlying, incomplete, or corrupted trajectories. IEEE Trans Pattern Anal Mach Intell 32(10):1832\u20131845","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"820_CR29","doi-asserted-by":"crossref","unstructured":"Tron R, Vidal R (2007) A benchmark for the comparison of 3-$$D$$ motion segmentation algorithms. In: Proceedings of the IEEE conference on computer vision and pattern recognition. IEEE, pp 1\u20138","DOI":"10.1109\/CVPR.2007.382974"},{"issue":"2","key":"820_CR30","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1109\/MSP.2010.939739","volume":"28","author":"R Vidal","year":"2011","unstructured":"Vidal R (2011) Subspace clustering. IEEE Sig Process Mag 28(2):52\u201368","journal-title":"IEEE Sig Process Mag"},{"key":"820_CR31","unstructured":"Wagstaff K, Cardie C, Rogers S, Schr\u00f6dl S (2001) Constrained $$k$$-means clustering with background knowledge. In: Proceedings of the 18th international conference on machine learning, vol\u00a01, pp 577\u2013584"},{"key":"820_CR32","unstructured":"Wang W, Carreira-Perpin\u00e1n MA (2013) Projection onto the probability simplex: an efficient algorithm with a simple proof, and an application."},{"key":"820_CR33","doi-asserted-by":"crossref","unstructured":"Wang X, Davidson I (2010) Flexible constrained spectral clustering. In: Proceedings of the 16th ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 563\u2013572","DOI":"10.1145\/1835804.1835877"},{"issue":"1","key":"820_CR34","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10618-012-0291-9","volume":"28","author":"X Wang","year":"2014","unstructured":"Wang X, Qian B, Davidson I (2014) On constrained spectral clustering and its applications. Data Min Knowl Disc 28(1):1\u201330","journal-title":"Data Min Knowl Disc"},{"key":"820_CR35","doi-asserted-by":"crossref","unstructured":"Yang J, Liang J, Wang K, Rosin P, Yang MH (2019) Subspace clustering via good neighbors. IEEE Trans Pattern Anal Mach Intell","DOI":"10.1109\/TPAMI.2019.2913863"},{"key":"820_CR36","doi-asserted-by":"crossref","unstructured":"Yeoh EJ, Ross ME, Shurtleff SA, Williams WK, Patel D, Mahfouz R, Behm FG, Raimondi SC, Relling MV, Patel A, Cheng C, Campana D, Wilkins D, Zhou X, Li J, HL H, Pui CH, Evans WE, Naeve C, Wong L, Downing JR (2002) Classification, subtype discovery, and prediction of outcome in pediatric acute lymphoblastic leukemia by gene expression profiling. Cancer Cell 1:133\u2013143","DOI":"10.1016\/S1535-6108(02)00032-6"},{"key":"820_CR37","doi-asserted-by":"crossref","unstructured":"You C, Robinson D, Vidal R (2016) Scalable sparse subspace clustering by orthogonal matching pursuit. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 3918\u20133927","DOI":"10.1109\/CVPR.2016.425"},{"issue":"2","key":"820_CR38","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 67(2):301\u2013320","journal-title":"J Roy Stat Soc Ser B"}],"container-title":["Data Mining and Knowledge Discovery"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10618-022-00820-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10618-022-00820-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10618-022-00820-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,5,16]],"date-time":"2022-05-16T11:28:36Z","timestamp":1652700516000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10618-022-00820-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,11]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,5]]}},"alternative-id":["820"],"URL":"https:\/\/doi.org\/10.1007\/s10618-022-00820-9","relation":{},"ISSN":["1384-5810","1573-756X"],"issn-type":[{"value":"1384-5810","type":"print"},{"value":"1573-756X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,11]]},"assertion":[{"value":"7 March 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 January 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 February 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}