We can think of matrices as tensors of order 2. Decomposing matrices is pretty well understood. I know little about decomposing tensors of rank higher than 2. But it is an active research area. For a flavour of the field, see maybe the tensorly decomposition example notebooks.
Fun applications listed in tntorch include
- Introduction
- Active Subspaces
- ANOVA Decomposition
- Arithmetics
- Automata
- Classification
- Tensor Completion
- Cross-approximation
- Tensor Decompositions
- Exponential Machines
- Boolean Logic
- Polynomial Chaos Expansions
- Sobol Indices
- Vector Fields
References
Anandkumar, Anima, Rong Ge, Daniel Hsu, Sham M. Kakade, and Matus Telgarsky. 2015. βTensor Decompositions for Learning Latent Variable Models (A Survey for ALT).β In Algorithmic Learning Theory, edited by Kamalika Chaudhuri, Claudio Gentile, and Sandra Zilles, 19β38. Lecture Notes in Computer Science. Springer International Publishing.
Anandkumar, Animashree, Rong Ge, Daniel Hsu, Sham M. Kakade, and Matus Telgarsky. 2014. βTensor Decompositions for Learning Latent Variable Models.β The Journal of Machine Learning Research 15 (1): 2773β2832.
Belkin, Mikhail, Luis Rademacher, and James Voss. 2016. βBasis Learning as an Algorithmic Primitive.β In Journal of Machine Learning Research, 446β87.
Bi, Xuan, Xiwei Tang, Yubai Yuan, Yanqing Zhang, and Annie Qu. 2021. βTensors in Statistics.β Annual Review of Statistics and Its Application 8 (1): 345β68.
Kossaifi, Jean, Yannis Panagakis, Anima Anandkumar, and Maja Pantic. 2019. βTensorLy: Tensor Learning in Python.β Journal of Machine Learning Research 20 (26): 1β6.
Rabusseau, Guillaume, and FranΓ§ois Denis. 2014. βLearning Negative Mixture Models by Tensor Decompositions.β arXiv:1403.4224 [Cs], March.
Robeva, E. 2016. βOrthogonal Decomposition of Symmetric Tensors.β SIAM Journal on Matrix Analysis and Applications 37 (1): 86β102.
Robeva, Elina, and Anna Seigal. 2016. βSingular Vectors of Orthogonally Decomposable Tensors.β arXiv:1603.09004 [Math], March.
Tenenbaum, J. B., and W. T. Freeman. 2000. βSeparating Style and Content with Bilinear Models.β Neural Computation 12 (6): 1247β83.
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