Editing Dimensionality reduction (section)

===Non-negative matrix factorization (NMF)===
{{Main|Non-negative matrix factorization}}

NMF decomposes a non-negative matrix to the product of two non-negative ones, which has been a promising tool in fields where only non-negative signals exist,<ref name="lee-seung">{{cite journal
 |author=Daniel D. Lee
 |author2=H. Sebastian Seung
 |author2-link=Sebastian Seung
 |name-list-style=amp
 |year=1999
 |title=Learning the parts of objects by non-negative matrix factorization
 |journal=[[Nature (journal)|Nature]]
 |volume=401
 |issue=6755
 |pages=788–791
 |doi=10.1038/44565
 |pmid=10548103
 |bibcode=1999Natur.401..788L
 |s2cid=4428232
 }}</ref><ref name="lee2001algorithms">{{cite conference
 |author1=Daniel D. Lee |author2=H. Sebastian Seung
 |name-list-style=amp |year=2001
 |url=https://proceedings.neurips.cc/paper/2000/file/f9d1152547c0bde01830b7e8bd60024c-Paper.pdf
 |title=Algorithms for Non-negative Matrix Factorization
 |conference=Advances in Neural Information Processing Systems 13: Proceedings of the 2000 Conference
 |pages=556–562
 |publisher=[[MIT Press]]
}}</ref> such as astronomy.<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><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 |doi-access=free }}</ref> NMF is well known since the multiplicative update rule by Lee & Seung,<ref name="lee-seung"/> which has been continuously developed: the inclusion of uncertainties,<ref name="blantonRoweis07"/> the consideration of missing data and parallel computation,<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> sequential construction<ref name="zhu16"/> which leads to the stability and linearity of NMF,<ref name="ren18"/> as well as other [[non-negative matrix factorization|updates]] including handling missing data in [[digital image processing]].<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 |doi-access=free }}</ref>

With a stable component basis during construction, and a linear modeling process, [[non-negative matrix factorization#Sequential NMF|sequential NMF]]<ref name="zhu16"/> is able to preserve the flux in direct imaging of circumstellar structures in astronomy,<ref name="ren18"/> as one of the [[methods of detecting exoplanets]], especially for the direct imaging of [[circumstellar disc]]s. In comparison with PCA, NMF does not remove the mean of the matrices, which leads to physical non-negative fluxes; therefore NMF is able to preserve more information than PCA as demonstrated by Ren et al.<ref name="ren18"/>