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Tensor decomposition
Any scheme for expressing a tensor as a sequence of elementary operations acting on other, often simpler tensors

In multilinear algebra, a tensor decomposition 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 decompositions.

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. The main tensor decompositions are:

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Notation

This section introduces basic notations and operations that are widely used in the field.

Table of symbols and their description.
SymbolsDefinition
a , a , a T , A , A {\displaystyle {a,{\bf {a}},{\bf {a}}^{T},\mathbf {A} ,{\mathcal {A}}}} scalar, vector, row, matrix, tensor
a = v e c ( . ) {\displaystyle {\bf {a}}={vec(.)}} vectorizing either a matrix or a tensor
A [ m ] {\displaystyle {\bf {A}}_{[m]}} matrixized tensor A {\displaystyle {\mathcal {A}}}
× m {\displaystyle \times _{m}} mode-m product

Introduction

A multi-way graph with K perspectives is a collection of K matrices X 1 , X 2 . . . . . X K {\displaystyle {X_{1},X_{2}.....X_{K}}} with dimensions I × J (where I, J are the number of nodes). This collection of matrices is naturally represented as a tensor X of size I × J × K. In order to avoid overloading the term “dimension”, we call an I × J × K tensor a three “mode” tensor, where “modes” are the numbers of indices used to index the tensor.

References

  1. Vasilescu, MAO; Terzopoulos, D (2007). "Multilinear (tensor) image synthesis, analysis, and recognition [exploratory dsp]". IEEE Signal Processing Magazine. 24 (6): 118–123. Bibcode:2007ISPM...24R.118V. doi:10.1109/MSP.2007.906024. /wiki/Bibcode_(identifier) ↩

  2. Kolda, Tamara G.; Bader, Brett W. (2009-08-06). "Tensor Decompositions and Applications". SIAM Review. 51 (3): 455–500. Bibcode:2009SIAMR..51..455K. doi:10.1137/07070111X. ISSN 0036-1445. S2CID 16074195. http://epubs.siam.org/doi/10.1137/07070111X ↩

  3. Sidiropoulos, Nicholas D.; De Lathauwer, Lieven; Fu, Xiao; Huang, Kejun; Papalexakis, Evangelos E.; Faloutsos, Christos (2017-07-01). "Tensor Decomposition for Signal Processing and Machine Learning". IEEE Transactions on Signal Processing. 65 (13): 3551–3582. arXiv:1607.01668. Bibcode:2017ITSP...65.3551S. doi:10.1109/TSP.2017.2690524. ISSN 1053-587X. S2CID 16321768. https://ieeexplore.ieee.org/document/7891546 ↩

  4. Bernardi, A.; Brachat, J.; Comon, P.; Mourrain, B. (2013-05-01). "General tensor decomposition, moment matrices and applications". Journal of Symbolic Computation. 52: 51–71. arXiv:1105.1229. doi:10.1016/j.jsc.2012.05.012. ISSN 0747-7171. S2CID 14181289. https://www.sciencedirect.com/science/article/pii/S0747717112001290 ↩

  5. Vasilescu, MAO; Terzopoulos, D (2007). "Multilinear (tensor) image synthesis, analysis, and recognition [exploratory dsp]". IEEE Signal Processing Magazine. 24 (6): 118–123. Bibcode:2007ISPM...24R.118V. doi:10.1109/MSP.2007.906024. /wiki/Bibcode_(identifier) ↩

  6. Rabanser, Stephan; Shchur, Oleksandr; Günnemann, Stephan (2017). "Introduction to Tensor Decompositions and their Applications in Machine Learning". arXiv:1711.10781 [stat.ML]. /wiki/ArXiv_(identifier) ↩

  7. Papalexakis, Evangelos E. (2016-06-30). "Automatic Unsupervised Tensor Mining with Quality Assessment". Proceedings of the 2016 SIAM International Conference on Data Mining. Society for Industrial and Applied Mathematics. pp. 711–719. arXiv:1503.03355. doi:10.1137/1.9781611974348.80. ISBN 978-1-61197-434-8. S2CID 10147789. 978-1-61197-434-8 ↩

  8. Vasilescu, M.A.O.; Terzopoulos, D. (2002). Multilinear Analysis of Image Ensembles: TensorFaces (PDF). Lecture Notes in Computer Science; (Presented at Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark). Vol. 2350. Springer, Berlin, Heidelberg. doi:10.1007/3-540-47969-4_30. ISBN 978-3-540-43745-1. 978-3-540-43745-1 ↩

  9. Gujral, Ekta; Pasricha, Ravdeep; Papalexakis, Evangelos E. (7 May 2018). Ester, Martin; Pedreschi, Dino (eds.). Proceedings of the 2018 SIAM International Conference on Data Mining. doi:10.1137/1.9781611975321. hdl:10536/DRO/DU:30109588. ISBN 978-1-61197-532-1. S2CID 21674935. 978-1-61197-532-1 ↩

  10. Gujral, Ekta; Papalexakis, Evangelos E. (9 October 2020). "OnlineBTD: Streaming Algorithms to Track the Block Term Decomposition of Large Tensors". 2020 IEEE 7th International Conference on Data Science and Advanced Analytics (DSAA). pp. 168–177. doi:10.1109/DSAA49011.2020.00029. ISBN 978-1-7281-8206-3. S2CID 227123356. 978-1-7281-8206-3 ↩

  11. Gujral, Ekta (2022). "Modeling and Mining Multi-Aspect Graphs With Scalable Streaming Tensor Decomposition". arXiv:2210.04404 [cs.SI]. /wiki/ArXiv_(identifier) ↩

  12. Vasilescu, M.A.O.; Kim, E. (2019). Compositional Hierarchical Tensor Factorization: Representing Hierarchical Intrinsic and Extrinsic Causal Factors. In The 25th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD’19): Tensor Methods for Emerging Data Science Challenges. arXiv:1911.04180. /wiki/ArXiv_(identifier) ↩

  13. De Lathauwer, Lieven (2008). "Decompositions of a Higher-Order Tensor in Block Terms—Part II: Definitions and Uniqueness". SIAM Journal on Matrix Analysis and Applications. 30 (3): 1033–1066. doi:10.1137/070690729. http://epubs.siam.org/doi/10.1137/070690729 ↩

  14. Vasilescu, M.A.O.; Kim, E.; Zeng, X.S. (2021), "CausalX: Causal eXplanations and Block Multilinear Factor Analysis", Conference Proc. of the 2020 25th International Conference on Pattern Recognition (ICPR 2020), pp. 10736–10743, arXiv:2102.12853, doi:10.1109/ICPR48806.2021.9412780, ISBN 978-1-7281-8808-9, S2CID 232046205 978-1-7281-8808-9 ↩

  15. Vasilescu, M.A.O.; Kim, E. (2019). Compositional Hierarchical Tensor Factorization: Representing Hierarchical Intrinsic and Extrinsic Causal Factors. In The 25th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD’19): Tensor Methods for Emerging Data Science Challenges. arXiv:1911.04180. /wiki/ArXiv_(identifier) ↩

  16. Gujral, Ekta; Pasricha, Ravdeep; Papalexakis, Evangelos (2020-04-20). "Beyond Rank-1: Discovering Rich Community Structure in Multi-Aspect Graphs". Proceedings of the Web Conference 2020. Taipei Taiwan: ACM. pp. 452–462. doi:10.1145/3366423.3380129. ISBN 978-1-4503-7023-3. S2CID 212745714. 978-1-4503-7023-3 ↩