Abstract
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.
Keywords
Multilinear algebraTensor (intrinsic definition)Singular value decompositionTensor algebraMultilinear mapToolboxPrincipal component analysisRank (graph theory)MathematicsAlgebra over a fieldCartesian tensorLinear algebraSoftwareMatrix (chemical analysis)Tucker decompositionTensor decompositionComputer sciencePure mathematicsTensor densityAlgorithmCombinatoricsMathematical analysisGeometryExact solutions in general relativityTensor fieldStatistics
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Publication Info
- Year
- 2009
- Type
- article
- Volume
- 51
- Issue
- 3
- Pages
- 455-500
- Citations
- 9981
- Access
- Closed
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Cite This
Tamara G. Kolda,
Brett W. Bader
(2009).
Tensor Decompositions and Applications.
SIAM Review
, 51
(3)
, 455-500.
https://doi.org/10.1137/07070111x
Identifiers
- DOI
- 10.1137/07070111x