In [[multilinear algebra]], a '''tensor decomposition''' 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:
* [[tensor rank decomposition]];
* [[tensor 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>;
* [[higher-order singular value decomposition]];
* [[Tucker decomposition]];
* [[matrix product state]]s, and operators or tensor trains;
* [[Online Tensor Decompositions]]<ref>{{cite web |last1=Gujral |first1=Ekta |title=Modeling and Mining Multi-Aspect Graphs With Scalable Streaming Tensor Decomposition |url=https://arxiv.org/abs/2210.04404}}</ref><ref>{{cite web |last1=Gujral |first1=Ekta |title=OnlineBTD: Streaming Algorithms to Track the Block Term Decomposition of Large Tensors |url=https://ieeexplore.ieee.org/abstract/document/9260061 |website=IEEE |publisher=WWW '20: Proceedings of The Web Conference 2020}}</ref>;
* [[hierarchical Tucker decomposition]]; and
* [[block term decomposition]].
* [[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>.
