{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,8]],"date-time":"2026-08-08T17:15:38Z","timestamp":1786209338059,"version":"3.56.0"},"reference-count":46,"publisher":"MIT Press - Journals","issue":"9","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2011,9]]},"abstract":"<jats:p> This letter describes algorithms for nonnegative matrix factorization (NMF) with the \u03b2-divergence (\u03b2-NMF). The \u03b2-divergence is a family of cost functions parameterized by a single shape parameter \u03b2 that takes the Euclidean distance, the Kullback-Leibler divergence, and the Itakura-Saito divergence as special cases (\u03b2 = 2, 1, 0 respectively). The proposed algorithms are based on a surrogate auxiliary function (a local majorization of the criterion function). We first describe a majorization-minimization algorithm that leads to multiplicative updates, which differ from standard heuristic multiplicative updates by a \u03b2-dependent power exponent. The monotonicity of the heuristic algorithm can, however, be proven for \u03b2 \u2208 (0, 1) using the proposed auxiliary function. Then we introduce the concept of the majorization-equalization (ME) algorithm, which produces updates that move along constant level sets of the auxiliary function and lead to larger steps than MM. Simulations on synthetic and real data illustrate the faster convergence of the ME approach. The letter also describes how the proposed algorithms can be adapted to two common variants of NMF: penalized NMF (when a penalty function of the factors is added to the criterion function) and convex NMF (when the dictionary is assumed to belong to a known subspace). <\/jats:p>","DOI":"10.1162\/neco_a_00168","type":"journal-article","created":{"date-parts":[[2011,6,15]],"date-time":"2011-06-15T10:10:26Z","timestamp":1308132626000},"page":"2421-2456","source":"Crossref","is-referenced-by-count":538,"title":["Algorithms for Nonnegative Matrix Factorization with the \u03b2-Divergence"],"prefix":"10.1162","volume":"23","author":[{"given":"C\u00e9dric","family":"F\u00e9votte","sequence":"first","affiliation":[{"name":"Laboratoire Traitement et Communication de l'Information (CNRS and T\u00e9l\u00e9com Paris Tech), 75014 Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J\u00e9r\u00f4me","family":"Idier","sequence":"additional","affiliation":[{"name":"Institute de Recherche en Communications et Cybernetique de Nantes (CNRS, Ecole Centrale de Nantes, Ecole des Mines de Nantes, and Universit\u00e9 de Nantes), 44000 Nantes, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1109\/TNN.2010.2076831"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/85.3.549"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2006.11.006"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1109\/ICASSP.2009.4959891"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0308531101"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1109\/83.743861"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.3390\/e12061532"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1016\/j.patrec.2008.02.016"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1007\/11679363_5"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.1986.4307748"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1109\/42.232263"},{"key":"B13","volume-title":"Proc. 11th International Society for Music Information Retrieval Conference","author":"Dessein A.","year":"2010"},{"key":"B14","volume-title":"Advances in neural information processing systems","volume":"19","author":"Dhillon I. 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