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Tensor decompositions and applications. (English) Zbl 1173.65029

Summary: 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:

65F20 Numerical solutions to overdetermined systems, pseudoinverses
15A69 Multilinear algebra, tensor calculus
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65Y15 Packaged methods for numerical algorithms
62H25 Factor analysis and principal components; correspondence analysis
65C60 Computational problems in statistics (MSC2010)
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