Tensor decomposition

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Short description: Process in algebra


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:

Notation

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

Table of symbols and their description.
Symbols Definition
a,𝐚,𝐚T,𝐀,𝒜 scalar, vector, row, matrix, tensor
𝐚=vec(.) vectorizing either a matrix or a tensor
𝐀[m] matrixized tensor 𝒜
×m mode-m product

Introduction

A multi-way graph with K perspectives is a collection of K matrices X1,X2.....XK 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. ↑ 1.0 1.1 Vasilescu, MAO; Terzopoulos, D (2007). "Multilinear (tensor) image synthesis, analysis, and recognition [exploratory dsp]". IEEE Signal Processing Magazine 24 (6): 118–123. doi:10.1109/MSP.2007.906024. Bibcode: 2007ISPM...24R.118V. 
  2. ↑ Kolda, Tamara G.; Bader, Brett W. (2009-08-06). "Tensor Decompositions and Applications" (in en). SIAM Review 51 (3): 455–500. doi:10.1137/07070111X. ISSN 0036-1445. Bibcode: 2009SIAMR..51..455K. 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. doi:10.1109/TSP.2017.2690524. ISSN 1053-587X. Bibcode: 2017ITSP...65.3551S. 
  4. ↑ Bernardi, A.; Brachat, J.; Comon, P.; Mourrain, B. (2013-05-01). "General tensor decomposition, moment matrices and applications" (in en). Journal of Symbolic Computation 52: 51–71. doi:10.1016/j.jsc.2012.05.012. ISSN 0747-7171. https://www.sciencedirect.com/science/article/pii/S0747717112001290. 
  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" (in en). Proceedings of the 2016 SIAM International Conference on Data Mining. Society for Industrial and Applied Mathematics. pp. 711–719. doi:10.1137/1.9781611974348.80. ISBN 978-1-61197-434-8. https://epubs.siam.org/doi/10.1137/1.9781611974348.80. 
  7. ↑ Vasilescu, M.A.O.; Terzopoulos, D. (2002). Multilinear Analysis of Image Ensembles: TensorFaces. Lecture Notes in Computer Science; (Presented at Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark). 2350. Springer, Berlin, Heidelberg. doi:10.1007/3-540-47969-4_30. ISBN 978-3-540-43745-1. http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf. Retrieved 2023-03-19. 
  8. ↑ Gujral, Ekta; Pasricha, Ravdeep; Papalexakis, Evangelos E. (7 May 2018). Proceedings of the 2018 SIAM International Conference on Data Mining. doi:10.1137/1.9781611975321. ISBN 978-1-61197-532-1. 
  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. 
  10. ↑ Gujral, Ekta (2022). "Modeling and Mining Multi-Aspect Graphs With Scalable Streaming Tensor Decomposition". arXiv:2210.04404 [cs.SI].
  11. ↑ 11.0 11.1 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. 
  12. ↑ De Lathauwer, Lieven (2008). "Decompositions of a Higher-Order Tensor in Block Terms—Part II: Definitions and Uniqueness" (in en). SIAM Journal on Matrix Analysis and Applications 30 (3): 1033–1066. doi:10.1137/070690729. http://epubs.siam.org/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, doi:10.1109/ICPR48806.2021.9412780, ISBN 978-1-7281-8808-9 
  14. ↑ Gujral, Ekta; Pasricha, Ravdeep; Papalexakis, Evangelos (2020-04-20). "Beyond Rank-1: Discovering Rich Community Structure in Multi-Aspect Graphs" (in en). Proceedings of the Web Conference 2020. Taipei Taiwan: ACM. pp. 452–462. doi:10.1145/3366423.3380129. ISBN 978-1-4503-7023-3. https://dl.acm.org/doi/10.1145/3366423.3380129. 




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