2009•SIAM ReviewRequires access

Tensor Decompositions and Applications

Tamara G. Kolda, Brett W. Bader

Abstract

Abstract. This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or [Formula: see text]-way array. Decompositions of higher-order tensors (i.e., [Formula: see text]-way arrays with [Formula: see text]) have applications in psychometrics, 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.

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Key concepts: Multilinear algebra, Tensor (intrinsic definition), Singular value decomposition, Tensor algebra, Multilinear map, Toolbox, Principal component analysis, Rank (graph theory)