Non-negative matrix factorization: Difference between revisions

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If we furthermore impose an orthogonality constraint on <math> \mathbf{H} </math>,

i.e. <math> \mathbf{H}\mathbf{H}^T = I </math>, then the above minimization is mathematically equivalent to the minimization of [[K-means clustering]] .<ref name="DingSDM2005" />.

Furthermore, the computed <math> H </math> gives the cluster membership, i.e.,

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|citeseerx=10.1.1.137.8281

}}</ref>

# Scalability: how to factorize million-by-billion matrices, which are commonplace in Web-scale data mining, e.g., see Distributed Nonnegative Matrix Factorization (DNMF),<ref>{{Cite journal

|author1=Chao Liu |author2=Hung-chih Yang |author3=Jinliang Fan |author4=Li-Wei He |author5=Yi-Min Wang |name-list-style=amp | title = Distributed Nonnegative Matrix Factorization for Web-Scale Dyadic Data Analysis on MapReduce

| journal = Proceedings of the 19th International World Wide Web Conference

| year = 2010

| url = http://research.microsoft.com/pubs/119077/DNMF.pdf

}}</ref>, Scalable Nonnegative Matrix Factorization (ScalableNMF),<ref>{{Cite journal

| author = Jiangtao Yin

| author2 = Lixin Gao

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| year = 2014

| url = http://rio.ecs.umass.edu/mnilpub/papers/ecmlpkdd2014-yin.pdf

}}</ref>, Distributed Stochastic Singular Value Decomposition.<ref>{{Cite web|url=https://mahout.apache.org/|title=Apache Mahout|website=mahout.apache.org|access-date=2019-12-14}}</ref>

# Online: how to update the factorization when new data comes in without recomputing from scratch, e.g., see online CNSC<ref>{{Cite journal |author1=Dong Wang |author2=Ravichander Vipperla |author3=Nick Evans |author4=Thomas Fang Zheng |title=Online Non-Negative Convolutive Pattern Learning for Speech Signals |journal=IEEE Transactions on Signal Processing |year=2013 |url=http://cslt.riit.tsinghua.edu.cn:8081/homepages/wangd/public/pdf/cnsc-tsp.pdf |doi=10.1109/tsp.2012.2222381 |volume=61 |issue=1 |pages=44–56 |bibcode=2013ITSP...61...44W |citeseerx=10.1.1.707.7348 |s2cid=12530378 |access-date=2015-04-19 |archive-url=https://web.archive.org/web/20150419072552/http://cslt.riit.tsinghua.edu.cn:8081/homepages/wangd/public/pdf/cnsc-tsp.pdf |archive-date=2015-04-19 |url-status=dead }}</ref>

# Collective (joint) factorization: factorizing multiple interrelated matrices for multiple-view learning, e.g. multi-view clustering, see CoNMF<ref>{{Cite journal