Tensor decomposition

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.^[1]^[2]^[3] Many tensor decompositions generalize some matrix decompositions.^[4]

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.^[1]^[5] The main tensor decompositions are:

Tensor rank decomposition;^[6]
Higher-order singular value decomposition;^[7]
Tucker decomposition;
matrix product states, and operators or tensor trains;
Online Tensor Decompositions^[8]^[9]^[10]
hierarchical Tucker decomposition;^[11]
block term decomposition^[12]^[13]^[11]^[14]

Notation

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

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

Introduction

A multi-way graph with K perspectives is a collection of K matrices ${X_{1},X_{2}.....X_{K}}$ 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

^ ^a ^b 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.
^ 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.
^ 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.
^ 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.
^ Rabanser, Stephan; Shchur, Oleksandr; Günnemann, Stephan (2017). "Introduction to Tensor Decompositions and their Applications in Machine Learning". arXiv:1711.10781 [stat.ML].
^ 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.
^ 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.
^ 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.
^ 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.
^ Gujral, Ekta (2022). "Modeling and Mining Multi-Aspect Graphs With Scalable Streaming Tensor Decomposition". arXiv:2210.04404 [cs.SI].
^ ^a ^b 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.
^ 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.
^ 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
^ 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.

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[VasilescuDSP-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.

[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.

[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.

[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.

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

[6] 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.

[7] 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.

[8] 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.

[9] 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.

[ektagujral-10] Gujral, Ekta (2022). "Modeling and Mining Multi-Aspect Graphs With Scalable Streaming Tensor Decomposition". arXiv:2210.04404 [cs.SI].

[Vasilescu2019-11] 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.

[12] 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.

[13] 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

[14] 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.

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