Tensor decomposition: Difference between revisions

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In [[multilinear algebra]], a '''tensor decomposition''' is any scheme for expressing a [[Tensor (machine learning)|"data tensor"]] (M-way array) as a sequence of elementary operations acting on other, often simpler tensors.<ref name=VasilescuDSP>{{cite journal|first1=MAO|last1=Vasilescu|first2=D|last2=Terzopoulos|title=Multilinear (tensor) image synthesis, analysis, and recognition [exploratory dsp]|journal=IEEE Signal Processing Magazine|date=2007 |volume=24|issue=6|pages=118–123|doi=10.1109/MSP.2007.906024 |bibcode=2007ISPM...24R.118V }}</ref><ref>{{Cite journal |last1=Kolda |first1=Tamara G. |last2=Bader |first2=Brett W. |date=2009-08-06 |title=Tensor Decompositions and Applications |url=http://epubs.siam.org/doi/10.1137/07070111X |journal=SIAM Review |language=en |volume=51 |issue=3 |pages=455–500 |doi=10.1137/07070111X |bibcode=2009SIAMR..51..455K |s2cid=16074195 |issn=0036-1445|url-access=subscription }}</ref><ref>{{Cite journal |last1=Sidiropoulos |first1=Nicholas D. |last2=De Lathauwer |first2=Lieven |last3=Fu |first3=Xiao |last4=Huang |first4=Kejun |last5=Papalexakis |first5=Evangelos E. |last6=Faloutsos |first6=Christos |date=2017-07-01 |title=Tensor Decomposition for Signal Processing and Machine Learning |url=https://ieeexplore.ieee.org/document/7891546 |journal=IEEE Transactions on Signal Processing |volume=65 |issue=13 |pages=3551–3582 |doi=10.1109/TSP.2017.2690524 |arxiv=1607.01668 |bibcode=2017ITSP...65.3551S |s2cid=16321768 |issn=1053-587X}}</ref> Many tensor decompositions generalize some [[matrix decomposition]]s.<ref>{{Cite journal|date=2013-05-01|title=General tensor decomposition, moment matrices and applications|url=https://www.sciencedirect.com/science/article/pii/S0747717112001290|journal=Journal of Symbolic Computation|language=en|volume=52|pages=51–71|doi=10.1016/j.jsc.2012.05.012|issn=0747-7171|arxiv=1105.1229|last1=Bernardi |first1=A. |last2=Brachat |first2=J. |last3=Comon |first3=P. |last4=Mourrain |first4=B. |hdl=11572/134905 |s2cid=14181289 }}</ref>

[[Tensors]] are generalizations of matrices to higher dimensions (or rather to higher orders, i.e. the higher number of dimensions) and can consequently be treated as multidimensional fields.<ref name="VasilescuDSP"/><ref>{{Cite arXiv |last1=Rabanser |first1=Stephan |last2=Shchur |first2=Oleksandr |last3=Günnemann |first3=Stephan |date=2017 |title=Introduction to Tensor Decompositions and their Applications in Machine Learning |class=stat.ML |eprint=1711.10781}}</ref>

The main tensor decompositions are:

* [[Tensor rank decomposition]];<ref>{{Cite book |last=Papalexakis |first=Evangelos E. |chapter=Automatic Unsupervised Tensor Mining with Quality Assessment |date=2016-06-30 |title=Proceedings of the 2016 SIAM International Conference on Data Mining |chapter-url=https://epubs.siam.org/doi/10.1137/1.9781611974348.80 |language=en |publisher=Society for Industrial and Applied Mathematics |pages=711–719 |doi=10.1137/1.9781611974348.80 |arxiv=1503.03355 |isbn=978-1-61197-434-8|s2cid=10147789 }}</ref>

* [[Higher-order singular value decomposition]];<ref >{{Cite book
|first1 = M.A.O.
|last1 = Vasilescu

|first2 = D.|last2=Terzopoulos

|last2 = Terzopoulos

|url = http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf

|title=Multilinear Analysis of Image Ensembles: TensorFaces

|title = Multilinear Analysis of Image Ensembles: TensorFaces
|series = Lecture Notes in Computer Science; (Presented at Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark)

|publisher = Springer, Berlin, Heidelberg

|volume = 2350

|doi = 10.1007/3-540-47969-4_30

|isbn = 978-3-540-43745-1

|year = 2002

|archive-date = 2022-12-29

}}</ref>

|access-date = 2023-03-19

|archive-url = https://web.archive.org/web/20221229090931/http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf

|url-status = dead

}}</ref>

* [[Tucker decomposition]];

* [[matrix product state]]s, and operators or tensor trains;