{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/algorithms-for-nonnegative-matrix","title":"Algorithms for nonnegative matrix factorization with the beta-divergence","arxiv_id":"1010.1763","date":"2010-10-08","proceeding":null,"authors":["Cédric Févotte","Jérôme Idier"],"abstract":"This paper describes algorithms for nonnegative matrix factorization (NMF)\nwith the beta-divergence (beta-NMF). The beta-divergence is a family of cost\nfunctions parametrized by a single shape parameter beta that takes the\nEuclidean distance, the Kullback-Leibler divergence and the Itakura-Saito\ndivergence as special cases (beta = 2,1,0, respectively). The proposed\nalgorithms are based on a surrogate auxiliary function (a local majorization of\nthe criterion function). We first describe a majorization-minimization (MM)\nalgorithm that leads to multiplicative updates, which differ from standard\nheuristic multiplicative updates by a beta-dependent power exponent. The\nmonotonicity of the heuristic algorithm can however be proven for beta in (0,1)\nusing the proposed auxiliary function. Then we introduce the concept of\nmajorization-equalization (ME) algorithm which produces updates that move along\nconstant level sets of the auxiliary function and lead to larger steps than MM.\nSimulations on synthetic and real data illustrate the faster convergence of the\nME approach. The paper also describes how the proposed algorithms can be\nadapted to two common variants of NMF : penalized NMF (i.e., when a penalty\nfunction of the factors is added to the criterion function) and convex-NMF\n(when the dictionary is assumed to belong to a known subspace).","url_abs":"http://arxiv.org/abs/1010.1763v3","url_pdf":"http://arxiv.org/pdf/1010.1763v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"algorithms-for-nonnegative-matrix","repo_url":"https://github.com/neel-dey/robust-nmf","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1010.1763","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}