Tensor decomposition: Difference between revisions

Revision as of 08:45, 19 March 2023

Revision as of 10:05, 19 March 2023

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In [[multilinear algebra]], a '''tensor decomposition'''<ref name=VasilescuDSP>{{cite journal|first1=MAO|last1=Vasilescu|first2=D|last2=Terzopoulos|title=Multilinear (tensor) image synthesis, analysis, and recognition [exploratory dsp]|work=IEEE Signal Processing Magazine|volume=24|issue=(6)|pages=118-123}}</ref><ref>{{Cite journal |last=Kolda |first=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 |issn=0036-1445}}</ref> <ref>{{Cite journal |last=Sidiropoulos |first=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=http://ieeexplore.ieee.org/document/7891546/ |journal=IEEE Transactions on Signal Processing |volume=65 |issue=13 |pages=3551–3582 |doi=10.1109/TSP.2017.2690524 |issn=1053-587X}}</ref> is any scheme for expressing a "data tensor" (M-way array) 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 name="VasilescuDSP"/><ref>{{Cite journal |last=Rabanser |first=Stephan |last2=Shchur |first2=Oleksandr |last3=Günnemann |first3=Stephan |date=2017 |title=Introduction to Tensor Decompositions and their Applications in Machine Learning |url=https://arxiv.org/abs/1711.10781 |doi=10.48550/ARXIV.1711.10781}}</ref>.

The main tensor decompositions are:

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

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

|first2=D.|last2=Terzopoulos

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

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

|year=2002

}}</ref>;

* [[Tucker decomposition]];

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

* [[Online Tensor Decompositions]]<ref>{{Cite journal |last1=Gujral |first1=Ekta |last2=Pasricha |first2=Ravdeep |last3=Papalexakis |first3=Evangelos E. |title=SamBaTen: Sampling-based Batch Incremental Tensor Decomposition|journal=Proceedings of the 2018 SIAM International Conference on Data Mining |date=7 May 2018 |doi=10.1137/1.9781611975321}}</ref><ref>{{Cite journal |last1=Gujral |first1=Ekta |last2=Papalexakis |first2=Evangelos E. |title=OnlineBTD: Streaming Algorithms to Track the Block Term Decomposition of Large Tensors |journal=2020 IEEE 7th International Conference on Data Science and Advanced Analytics (DSAA) |date=9 October 2020 |doi=10.1109/DSAA49011.2020.00029}}</ref><ref name="ektagujral">{{Cite journal |last=Gujral |first=Ekta |date=2022 |title=Modeling and Mining Multi-Aspect Graphs With Scalable Streaming Tensor Decomposition |url=https://arxiv.org/abs/2210.04404 |doi=10.48550/ARXIV.2210.04404}}</ref>

* [[hierarchical Tucker decomposition]];<ref name=Vasilescu2019>{{Citation |first1=M.A.O.|last1=Vasilescu|first2=E.|last2=Kim|date=2019|title=Compositional Hierarchical Tensor Factorization: Representing Hierarchical Intrinsic and Extrinsic Causal Factors|work=In The 25th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD’19): Tensor Methods for Emerging Data Science Challenges|url=https://arxiv.org/pdf/1911.04180.pdf}}</ref>;

* [[block term decomposition]]<ref>{{Cite journal |last=De Lathauwer|first=Lieven |title=Decompositions of a Higher-Order Tensor in Block Terms—Part II: Definitions and Uniqueness |url=http://epubs.siam.org/doi/10.1137/070690729 |journal=SIAM Journal on Matrix Analysis and Applications |language=en |doi=10.1137/070690729}}</ref><ref>{{citation|first1=M.A.O.|last1=Vasilescu|first2=E.|last2=Kim|first3=X.S.|last3=Zeng|title=CausalX: Causal eXplanations and Block Multilinear Factor Analysis|work=Conference Proc. of the 2020 25th International Conference on Pattern Recognition (ICPR 2020)}}</ref><ref name="Vasilescu2019/><ref>{{cite journal |last=Gujral |first=Ekta |last2=Pasricha |first2=Ravdeep |last3=Papalexakis |first3=Evangelos |date=2020-04-20 |title=Beyond Rank-1: Discovering Rich Community Structure in Multi-Aspect Graphs |url=https://dl.acm.org/doi/10.1145/3366423.3380129 |journal=Proceedings of The Web Conference 2020 |language=en |___location=Taipei Taiwan |publisher=ACM |pages=452–462 |doi=10.1145/3366423.3380129 |isbn=978-1-4503-7023-3}}</ref>

==Preliminary Definitions and Notation==