Non-negative matrix factorization: Difference between revisions

Revision as of 20:09, 21 May 2023

Revision as of 17:35, 24 July 2023

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Many standard NMF algorithms analyze all the data together; i.e., the whole matrix is available from the start. This may be unsatisfactory in applications where there are too many data to fit into memory or where the data are provided in [[Data stream|streaming]] fashion. One such use is for [[collaborative filtering]] in [[recommender system|recommendation systems]], where there may be many users and many items to recommend, and it would be inefficient to recalculate everything when one user or one item is added to the system. The cost function for optimization in these cases may or may not be the same as for standard NMF, but the algorithms need to be rather different.<ref>{{cite book|url=http://dl.acm.org/citation.cfm?id=1339264.1339709|title=Online Discussion Participation Prediction Using Non-negative Matrix Factorization |first1=Yik-Hing|last1=Fung|first2=Chun-Hung|last2=Li|first3=William K.|last3=Cheung|date=2 November 2007|publisher=IEEE Computer Society|pages=284–287|via=dl.acm.org|isbn=9780769530284|series=Wi-Iatw '07}}</ref><ref>{{Cite journal |author=Naiyang Guan|author2=Dacheng Tao|author3=Zhigang Luo|author4=Bo Yuan|name-list-style=amp|date=July 2012|title=Online Nonnegative Matrix Factorization With Robust Stochastic Approximation|journal=IEEE Transactions on Neural Networks and Learning Systems |issue=7 |doi=10.1109/TNNLS.2012.2197827|pmid=24807135|volume=23|pages=1087–1099|s2cid=8755408}}</ref>

=== Convolutional NMF ===

If the columns of {{math|'''V'''}} represent data sampled over spatial or temporal dimensions, e.g. time signals, images, or video, features that are equivariant w.r.t. shifts along these dimensions can be learned by Convolutional NMF. In this case, {{math|'''W'''}} is sparse with columns having local non-zero weight windows that are shared across shifts along the spatio-temporal dimensions of {{math|'''V'''}}, representing [[Kernel (image processing)|convolution kernels]]. By spatio-temporal pooling of {{math|'''H'''}} and repeatedly using the resulting representation as input to convolutional NMF, deep feature hierarchies can be learned.<ref>{{Cite journalbook |last=Behnke |first=S. |datetitle=Proceedings of the International Joint Conference on Neural Networks, 2003 |titlechapter=Discovering hierarchical speech features using convolutional non-negative matrix factorization |date=2003 |chapter-url=https://ieeexplore.ieee.org/document/1224004 |journal=Proceedings of the International Joint Conference on Neural Networks, 2003. |___location=Portland, Oregon USA |publisher=IEEE |volume=4 |pages=2758–2763 |doi=10.1109/IJCNN.2003.1224004 |isbn=978-0-7803-7898-8|s2cid=3109867 }}</ref>

== Algorithms ==

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|url = <!-- http://www.cc.gatech.edu/~jingu/docs/2011_paper_sisc_nmf.pdf --><!-- removing dead link -->

|doi = 10.1137/110821172

|bibcode = 2011SJSC...33.3261K

|citeseerx = 10.1.1.419.798

}}</ref> among several others.<ref name="kim2013unified">{{Cite journal

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=== Astronomy ===

In astronomy, NMF is a promising method for [[dimension reduction]] in the sense that astrophysical signals are non-negative. NMF has been applied to the spectroscopic observations<ref name=":0">{{Cite journal |last1=Berné |first1=O. |last2=Joblin |first2=C.|author2-link=Christine Joblin |last3=Deville |first3=Y. |last4=Smith |first4=J. D. |last5=Rapacioli |first5=M. |last6=Bernard |first6=J. P. |last7=Thomas |first7=J. |last8=Reach |first8=W. |last9=Abergel |first9=A. |date=2007-07-01 |title=Analysis of the emission of very small dust particles from Spitzer spectro-imagery data using blind signal separation methods |url=https://www.aanda.org/articles/aa/abs/2007/26/aa6282-06/aa6282-06.html |journal=Astronomy & Astrophysics |language=en |volume=469 |issue=2 |pages=575–586 |doi=10.1051/0004-6361:20066282 |arxiv=astro-ph/0703072 |bibcode=2007A&A...469..575B |issn=0004-6361|doi-access=free }}</ref><ref name="blantonRoweis07">{{Cite journal |arxiv=astro-ph/0606170|last1= Blanton|first1= Michael R.|title= K-corrections and filter transformations in the ultraviolet, optical, and near infrared |journal= The Astronomical Journal|volume= 133|issue= 2|pages= 734–754|last2= Roweis|first2= Sam |year= 2007|doi= 10.1086/510127|bibcode = 2007AJ....133..734B |s2cid= 18561804}}</ref> and the direct imaging observations<ref name = "ren18">{{Cite journal|arxiv=1712.10317|last1= Ren|first1= Bin |title= Non-negative Matrix Factorization: Robust Extraction of Extended Structures|journal= The Astrophysical Journal|volume= 852|issue= 2|pages= 104|last2= Pueyo|first2= Laurent|last3= Zhu | first3 = Guangtun B.|last4= Duchêne|first4= Gaspard |year= 2018|doi= 10.3847/1538-4357/aaa1f2|bibcode = 2018ApJ...852..104R |s2cid= 3966513}}</ref> as a method to study the common properties of astronomical objects and post-process the astronomical observations. The advances in the spectroscopic observations by Blanton & Roweis (2007)<ref name="blantonRoweis07" /> takes into account of the uncertainties of astronomical observations, which is later improved by Zhu (2016)<ref name="zhu16">{{Cite arXiv|last=Zhu|first=Guangtun B.|date=2016-12-19|title=Nonnegative Matrix Factorization (NMF) with Heteroscedastic Uncertainties and Missing data |eprint=1612.06037|class=astro-ph.IM}}</ref> where missing data are also considered and [[parallel computing]] is enabled. Their method is then adopted by Ren et al. (2018)<ref name="ren18" /> to the direct imaging field as one of the [[methods of detecting exoplanets]], especially for the direct imaging of [[circumstellar disks]].

Ren et al. (2018)<ref name="ren18" /> are able to prove the stability of NMF components when they are constructed sequentially (i.e., one by one), which enables the [[linearity]] of the NMF modeling process; the [[linearity]] property is used to separate the stellar light and the light scattered from the [[exoplanets]] and [[circumstellar disks]].

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NMF has also been applied to citations data, with one example clustering [[English Wikipedia]] articles and [[scientific journal]]s based on the outbound scientific citations in English Wikipedia.<ref>{{Cite conference

| last1 = Nielsen

| firstfirst1 = Finn Årup

| title = Clustering of scientific citations in Wikipedia

| conference = [[Wikimania]]