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.

MSC codes

  1. 15A69
  2. 65F99

Keywords

  1. tensor decompositions
  2. multiway arrays
  3. multilinear algebra
  4. parallel factors (PARAFAC)
  5. canonical decomposition (CANDECOMP)
  6. higher-order principal components analysis (Tucker)
  7. higher-order singular value decomposition (HOSVD)

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Information & Authors

Information

Published In

cover image SIAM Review
SIAM Review
Pages: 455 - 500
ISSN (online): 1095-7200

History

Submitted: 24 August 2007
Accepted: 3 June 2008
Published online: 5 August 2009

MSC codes

  1. 15A69
  2. 65F99

Keywords

  1. tensor decompositions
  2. multiway arrays
  3. multilinear algebra
  4. parallel factors (PARAFAC)
  5. canonical decomposition (CANDECOMP)
  6. higher-order principal components analysis (Tucker)
  7. higher-order singular value decomposition (HOSVD)

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