{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T18:11:27Z","timestamp":1792087887324,"version":"4.4.0"},"reference-count":12,"publisher":"Wiley","license":[{"start":{"date-parts":[[2008,1,1]],"date-time":"2008-01-01T00:00:00Z","timestamp":1199145600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/100008393","name":"Danish Technical Research Council","doi-asserted-by":"crossref","award":["26-02-0092"],"award-info":[{"award-number":["26-02-0092"]}],"id":[{"id":"10.13039\/100008393","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100008393","name":"Danish Technical Research Council","doi-asserted-by":"crossref","award":["274-06-0521"],"award-info":[{"award-number":["274-06-0521"]}],"id":[{"id":"10.13039\/100008393","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Danish Research Council for Technology, and Production Sciences","award":["26-02-0092"],"award-info":[{"award-number":["26-02-0092"]}]},{"name":"Danish Research Council for Technology, and Production Sciences","award":["274-06-0521"],"award-info":[{"award-number":["274-06-0521"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computational Intelligence and Neuroscience"],"published-print":{"date-parts":[[2008]]},"abstract":"<jats:p>We investigate the conditions for which nonnegative matrix factorization (NMF) is unique and introduce several theorems which can determine whether the decomposition is in fact unique or not. The theorems are illustrated by several examples showing the use of the theorems and their limitations. We have shown that corruption of a unique NMF matrix by additive noise leads to a noisy estimation of the noise-free unique solution. Finally, we use a stochastic view of NMF to analyze which characterization of the underlying model will result in an NMF with small estimation errors.<\/jats:p>","DOI":"10.1155\/2008\/764206","type":"journal-article","created":{"date-parts":[[2008,5,15]],"date-time":"2008-05-15T10:19:33Z","timestamp":1210846773000},"page":"1-9","source":"Crossref","is-referenced-by-count":141,"title":["Theorems on Positive Data: On the Uniqueness of NMF"],"prefix":"10.1155","volume":"2008","author":[{"given":"Hans","family":"Laurberg","sequence":"first","affiliation":[{"name":"Department of Electronic Systems, Aalborg University, Niels Jernes Vej 12, 9220 Aalborg, Denmark"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mads Gr\u00e6sb\u00f8ll","family":"Christensen","sequence":"additional","affiliation":[{"name":"Department of Electronic Systems, Aalborg University, Niels Jernes Vej 12, 9220 Aalborg, Denmark"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark D.","family":"Plumbley","sequence":"additional","affiliation":[{"name":"Department of Electronic Engineering, Queen Mary, University of London, Mile End Road, London E1 4NS, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lars Kai","family":"Hansen","sequence":"additional","affiliation":[{"name":"Department of Informatics and Mathematical Modeling, Technical University of Denmark, Richard Petersens Plads, Building 321, 2800 Lyngby, Denmark"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S\u00f8ren Holdt","family":"Jensen","sequence":"additional","affiliation":[{"name":"Department of Electronic Systems, Aalborg University, Niels Jernes Vej 12, 9220 Aalborg, Denmark"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","reference":[{"key":"2","doi-asserted-by":"publisher","DOI":"10.1038\/44565"},{"key":"3","first-page":"556","volume-title":"Algorithms for non-negative matrix factorization","year":"2000"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2006.11.006"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1137\/1015076"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1137\/1016064"},{"key":"7","first-page":"1141","volume-title":"When does non-negative matrix factorization give a correct decomposition into parts?","year":"2004"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2002.800502"},{"key":"9","year":"1970"},{"key":"10","year":"1988"},{"key":"11","year":"1996"},{"key":"13","first-page":"1457","volume":"5","year":"2004","journal-title":"The Journal of Machine Learning Research"},{"key":"18","year":"1974"}],"container-title":["Computational Intelligence and Neuroscience"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/cin\/2008\/764206.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/cin\/2008\/764206.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/cin\/2008\/764206.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,8,11]],"date-time":"2016-08-11T12:13:17Z","timestamp":1470917597000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/cin\/2008\/764206\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008]]},"references-count":12,"alternative-id":["764206","764206"],"URL":"https:\/\/doi.org\/10.1155\/2008\/764206","relation":{},"ISSN":["1687-5265","1687-5273"],"issn-type":[{"value":"1687-5265","type":"print"},{"value":"1687-5273","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008]]}}}