Tensor Decompositions and Applications

TLDR

Higher‑order tensors (N‑way arrays) are employed across numerous scientific and engineering fields, and their decompositions—such as CP and Tucker—extend matrix techniques to multi‑dimensional data. This survey provides an overview of higher‑order tensor decompositions, their applications, and available software. The survey reviews key tensor decomposition methods and discusses software tools such as the N‑way Toolbox, Tensor Toolbox, and Multilinear Engine.

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.