SIAM Review


SIAM Rev., 51(3), 455–500. (46 pages)

Tensor Decompositions and Applications

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Publication Data

ISSN (print): 0036-1445
ISSN (online): 1095-7200
Publisher: Society for Industrial and Applied Mathematics
CODEN: siread
Tamara G. Kolda and Brett W. Bader
https://doi.org/10.1137/07070111X

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or N-way array. Decompositions of higher-order tensors (i.e., N-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

Copyright © 2009 Society for Industrial and Applied Mathematics
Permalink: https://doi.org/10.1137/07070111X

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(2018) A Fast and Adaptive Method for Determining $K_{1}$ , $K_{2}$ , and $K_{3}$ in the Tensor Decomposition-Based Anomaly Detection Algorithm. IEEE Geoscience and Remote Sensing Letters 15:1, 3-7. Crossref
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(2018) Nonlocal image denoising via adaptive tensor nuclear norm minimization. Neural Computing and Applications 29:1, 3-19. Crossref
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(2013) Optimization-Based Algorithms for Tensor Decompositions: Canonical Polyadic Decomposition, Decomposition in Rank-$(L_r,L_r,1)$ Terms, and a New Generalization. SIAM Journal on Optimization 23:2, 695-720. Abstract | PDF (1376 KB) 
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