{"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/the-why-and-how-of-nonnegative-matrix","title":"The Why and How of Nonnegative Matrix Factorization","arxiv_id":"1401.5226","date":"2014-01-21","proceeding":null,"authors":["Nicolas Gillis"],"abstract":"Nonnegative matrix factorization (NMF) has become a widely used tool for the\nanalysis of high-dimensional data as it automatically extracts sparse and\nmeaningful features from a set of nonnegative data vectors. We first illustrate\nthis property of NMF on three applications, in image processing, text mining\nand hyperspectral imaging --this is the why. Then we address the problem of\nsolving NMF, which is NP-hard in general. We review some standard NMF\nalgorithms, and also present a recent subclass of NMF problems, referred to as\nnear-separable NMF, that can be solved efficiently (that is, in polynomial\ntime), even in the presence of noise --this is the how. Finally, we briefly\ndescribe some problems in mathematics and computer science closely related to\nNMF via the nonnegative rank.","url_abs":"http://arxiv.org/abs/1401.5226v2","url_pdf":"http://arxiv.org/pdf/1401.5226v2.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":"the-why-and-how-of-nonnegative-matrix","repo_url":"https://github.com/ds-personalization/movielens-recommendations","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"the-why-and-how-of-nonnegative-matrix","repo_url":"https://github.com/nzhinusoftcm/review-on-collaborative-filtering","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}},{"paper_slug":"the-why-and-how-of-nonnegative-matrix","repo_url":"https://github.com/prki/mastersthesis","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1401.5226","atlas_url":"https://app.syntology.ai/?focus=1401.5226","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}