Tensor decomposition: Difference between revisions

Revision as of 17:28, 12 March 2023

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{{Short description|Procees in algebra}}

{{Refimprove|date=June 2021}}

In [[multilinear algebra]], a '''tensor decomposition''' <ref name="Sidiropoulos">{{cite web |last1=Sidiropoulos |first1=Nicholas D. |title=Tensor Decomposition for Signal Processing and Machine Learning |url=https://ieeexplore.ieee.org/abstract/document/7891546 |publisher=IEEE Transactions on Signal Processing}}</ref> is any scheme for expressing a [[tensor]] as a sequence of elementary operations acting on other, often simpler tensors. 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. |s2cid=14181289 }}</ref>

Tensors are generalizations of matrices to higher dimensions and can consequently be treated as multidimensional fields <ref>{{cite web |last1=Rabanser |first1=Stephan |title=Introduction to Tensor Decompositions and their Applications in Machine Learning |url=https://arxiv.org/pdf/1711.10781.pdf}}</ref>.

The main tensor decompositions are:

* [[tensorTensor rank decomposition]]<ref>{{cite web |last1=Papalexakis |first1=Evangelos E. |title=Automatic unsupervised tensor mining with quality assessment |url=https://epubs.siam.org/doi/abs/10.1137/1.9781611974348.80}}</ref>;

* [[higherHigher-order singular value decomposition]];

* [[Tucker decomposition]];

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

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* [[block term decomposition]]<ref>{{cite web |last1=Lathauwer |first1=Lieven De |title=Decompositions of a Higher-Order Tensor in Block Terms—Part II: Definitions and Uniqueness |url=https://epubs.siam.org/doi/abs/10.1137/070690729}}</ref><ref>{{cite web |last1=Gujral |first1=Ekta |title=Beyond rank-1: Discovering rich community structure in multi-aspect graphs |url=https://dl.acm.org/doi/abs/10.1145/3366423.3380129}}</ref>.

==Introduction==

==CP Decomposition==

==Tucker decomposition==

==Online Tensor Decomposition==

==Block Term Decomposition==

==References==

{{Reflist}}