Tensor decomposition
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In multilinear algebra, a tensor decomposition [1] is any scheme for expressing a tensor as a sequence of elementary operations acting on other, often simpler tensors. Many tensor decompositions generalize some matrix decompositions.[2]
Tensors are generalizations of matrices to higher dimensions and can consequently be treated as multidimensional fields [3]. The main tensor decompositions are:
- Tensor rank decomposition[4];
- Higher-order singular value decomposition;
- Tucker decomposition;
- matrix product states, and operators or tensor trains;
- Online Tensor Decompositions[5][6];
- hierarchical Tucker decomposition; and
- block term decomposition[7][8].
Introduction
CP Decomposition
Tucker decomposition
Online Tensor Decomposition
Block Term Decomposition
References
- ^ Sidiropoulos, Nicholas D. "Tensor Decomposition for Signal Processing and Machine Learning". IEEE Transactions on Signal Processing.
- ^ 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. "Introduction to Tensor Decompositions and their Applications in Machine Learning" (PDF).
- ^ Papalexakis, Evangelos E. "Automatic unsupervised tensor mining with quality assessment".
- ^ Gujral, Ekta. "Modeling and Mining Multi-Aspect Graphs With Scalable Streaming Tensor Decomposition".
- ^ Gujral, Ekta. "OnlineBTD: Streaming Algorithms to Track the Block Term Decomposition of Large Tensors". IEEE. WWW '20: Proceedings of The Web Conference 2020.
- ^ Lathauwer, Lieven De. "Decompositions of a Higher-Order Tensor in Block Terms—Part II: Definitions and Uniqueness".
- ^ Gujral, Ekta. "Beyond rank-1: Discovering rich community structure in multi-aspect graphs".
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