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. “Tensor Decompositions for Learning Latent Variable Models (A Survey for ALT).” In Algorithmic Learning Theory, edited by Kamalika Chaudhuri, Claudio Gentile, and Sandra Zilles, 19–38. Lecture Notes in Computer Science. Springer International Publishing.
Anandkumar, Animashree, Rong Ge, Daniel Hsu, Sham M. Kakade, and Matus Telgarsky. 2014. “Tensor Decompositions for Learning Latent Variable Models.” The Journal of Machine Learning Research 15 (1): 2773–2832.
Belkin, Mikhail, Luis Rademacher, and James Voss. 2016. “Basis Learning as an Algorithmic Primitive.” In Journal of Machine Learning Research, 446–87.
Bi, Xuan, Xiwei Tang, Yubai Yuan, Yanqing Zhang, and Annie Qu. 2021. “Tensors in Statistics.” Annual Review of Statistics and Its Application 8 (1): 345–68.
Kossaifi, Jean, Yannis Panagakis, Anima Anandkumar, and Maja Pantic. 2019. “TensorLy: Tensor Learning in Python.” Journal of Machine Learning Research 20 (26): 1–6.
Rabusseau, Guillaume, and François Denis. 2014. “Learning Negative Mixture Models by Tensor Decompositions.” arXiv:1403.4224 [Cs], March.
Robeva, E. 2016. “Orthogonal Decomposition of Symmetric Tensors.” SIAM Journal on Matrix Analysis and Applications 37 (1): 86–102.
Robeva, Elina, and Anna Seigal. 2016. “Singular Vectors of Orthogonally Decomposable Tensors.” arXiv:1603.09004 [Math], March.
Tenenbaum, J. B., and W. T. Freeman. 2000. “Separating Style and Content with Bilinear Models.” Neural Computation 12 (6): 1247–83.
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