Non-negative matrix factorization: Difference between revisions

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'''Non-negative matrix factorization''' ('''NMF''' or '''NNMF'''), also '''non-negative matrix approximation'''<ref name="dhillon"/><ref>{{cite journal|last1=Tandon|first1=Rashish|author2=Suvrit Sra|title=Sparse nonnegative matrix approximation: new formulations and algorithms|year=2010|series=TR|url=https://is.tuebingen.mpg.de/fileadmin/user_upload/files/publications/MPIK-TR-193_%5B0%5D.pdf}}</ref> is a group of [[algorithm]]s in [[multivariate analysis]] and [[linear algebra]] where a [[matrix (mathematics)|matrix]] {{math|'''V'''}} is [[Matrix decomposition|factorized]] into (usually) two matrices {{math|'''W'''}} and {{math|'''H'''}}, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms or muscular activity, non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically.
 
NMF finds applications in such fields as [[astronomy]],<ref name="blantonRoweis07"/><ref name="ren18"/> [[computer vision]], [[document clustering]],<ref name="dhillon" /> [[Imputation (statistics)|missing data imputation]],<ref name="ren20">{{Cite journal|arxiv=2001.00563|last1= Ren|first1= Bin |title= Using Data Imputation for Signal Separation in High Contrast Imaging|journal= The Astrophysical Journal|volume= 892|issue= 2|pages= 74|last2= Pueyo|first2= Laurent|last3= Chen | first3 = Christine|last4= Choquet|first4= Elodie |last5= Debes|first5= John H|last6= Duechene |first6= Gaspard|last7= Menard|first7=Francois|last8=Perrin|first8=Marshall D.|year= 2020|doi= 10.3847/1538-4357/ab7024 | bibcode = 2020ApJ...892...74R |s2cid= 209531731}}</ref> [[chemometrics]], [[audio signal processing]], [[recommender system]]s,<ref name="gemulla">{{cite conference |author=Rainer Gemulla |author2=Erik Nijkamp |author3=Peter J. Haas|author3-link= Peter J. Haas (computer scientist)|author4=Yannis Sismanis |title=Large-scale matrix factorization with distributed stochastic gradient descent |conference=Proc. ACM SIGKDD Int'l Conf. on Knowledge discovery and data mining |url=<!-- http://www.mpi-inf.mpg.de/~rgemulla/publications/rj10481rev.pdf --><!--removing dead link--> |year=2011 |pages=69–77 }}</ref><ref>{{cite conference |author=Yang Bao|title=TopicMF: Simultaneously Exploiting Ratings and Reviews for Recommendation |conference=AAAI |url=http://www.aaai.org/ocs/index.php/AAAI/AAAI14/paper/view/8273 |year=2014 |display-authors=etal}}</ref> and [[bioinformatics]].<ref>{{cite journal |author=Ben Murrell|title=Non-Negative Matrix Factorization for Learning Alignment-Specific Models of Protein Evolution |journal=PLOS ONE |volume=6 |issue=12 |year=2011 |pages=e28898|display-authors=etal|doi=10.1371/journal.pone.0028898 |pmid=22216138 |pmc=3245233 |bibcode=2011PLoSO...628898M |doi-access=free }}</ref>
 
== History ==
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although it may also still be referred to as NMF.<ref>{{Cite conference | last1 = Hsieh | first1 = C. J. | last2 = Dhillon | first2 = I. S. | doi = 10.1145/2020408.2020577 | title = Fast coordinate descent methods with variable selection for non-negative matrix factorization | conference = Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '11 | pages = 1064| year = 2011 | isbn = 9781450308137 | url = http://www.cs.utexas.edu/~cjhsieh/nmf_kdd11.pdf}}</ref>
 
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* Andrzej Cichocki, Morten Mrup, et al.: "Advances in Nonnegative Matrix and Tensor Factorization", Hindawi Publishing Corporation, {{ISBN|978-9774540455}} (2008).
* Andrzej Cichocki, Rafal Zdunek, Anh Huy Phan and Shun-ichi Amari: "Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation", Wiley, {{ISBN|978-0470746660}} (2009).