Tensor decomposition
In multilinear algebra, a tensor decomposition[1][2] [3] 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.[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
scalar, vector, row, matrix, tensor
vectorizing either a matrix or a tensor
matrixized tensor
mode-m product
Introduction
A multi-way graph with K perspectives is a collection of K matrices 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. "Multilinear (tensor) image synthesis, analysis, and
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(http://epubs.siam.org/doi/10.1137/07070111X). SIAM Review. 51 (3): 455–500.
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Evangelos E.; Faloutsos, Christos (2017-07-01). "Tensor Decomposition for Signal
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article/pii/S0747717112001290). Journal of Symbolic Computation. 52: 51–71.
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s://arxiv.org/abs/1711.10781) [stat.ML (https://arxiv.org/archive/stat.ML)].
6. Papalexakis, Evangelos E. (2016-06-30). "Automatic Unsupervised Tensor Mining with
Quality Assessment" (https://epubs.siam.org/doi/10.1137/1.9781611974348.80).
Proceedings of the 2016 SIAM International Conference on Data Mining. Society for
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3355). doi:10.1137/1.9781611974348.80 (https://doi.org/10.1137%2F1.9781611974348.80).
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7. Vasilescu, M.A.O.; Terzopoulos, D. (2002). Multilinear Analysis of Image Ensembles:
TensorFaces (http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_fil
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4_30 (https://doi.org/10.1007%2F3-540-47969-4_30). ISBN 978-3-540-43745-1.
8. Gujral, Ekta; Pasricha, Ravdeep; Papalexakis, Evangelos E. (7 May 2018). Ester, Martin;
Pedreschi, Dino (eds.). "SamBaTen: Sampling-based Batch Incremental Tensor
Decomposition". Proceedings of the 2018 SIAM International Conference on Data Mining.
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61197-532-1. S2CID 21674935 (https://api.semanticscholar.org/CorpusID:21674935).
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Algorithms to Track the Block Term Decomposition of Large Tensors". 2020 IEEE 7th
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Tensor Decomposition". arXiv:2210.04404 (https://arxiv.org/abs/2210.04404) [cs.SI (https://ar
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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 (https://arxiv.org/abs/1911.04180).
12. De Lathauwer, Lieven (2008). "Decompositions of a Higher-Order Tensor in Block Terms—
Part II: Definitions and Uniqueness" (http://epubs.siam.org/doi/10.1137/070690729). SIAM
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Multilinear Factor Analysis", Conference Proc. of the 2020 25th International Conference on
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