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 $N geq3$) 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.

@article{kolda2009tensor, author = {Kolda, Tamara G. and Bader, Brett W.}, title = {Tensor Decompositions and Applications}, journal = {SIAM Review}, publisher = {SIAM}, year = {2009}, volume = {51}, number = {3}, pages = {455--500}, url = {http://dx.doi.org/10.1137/07070111X}, doi = {10.1137/07070111X}, issn = {00361445}, keywords = {decomposition, hosvd, parafac, tensor, three, tucker}, 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 $N geq3$) 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.} }

%0 = article %A = Kolda, Tamara G. and Bader, Brett W. %D = 2009 %I = SIAM %T = Tensor Decompositions and Applications %U = http://dx.doi.org/10.1137/07070111X

@incollection{hl84parafac, author = {Harshman, R. A. and Lundy, M. E.}, title = {The PARAFAC model for three-way factor analysis and multidimensional scaling}, editor = {Law, H. G. and Snyder Jr, C. W. and Hattie, J. A. and McDonald, R. P.}, booktitle = {Research methods for multimode data analysis}, publisher = {Praeger}, address = {New York}, year = {1984}, pages = {122--215}, keywords = {mode, parafac, three, 3mode, analysis} }

%0 = incollection %A = Harshman, R. A. and Lundy, M. E. %B = Research methods for multimode data analysis %C = New York %D = 1984 %I = Praeger %T = The PARAFAC model for three-way factor analysis and multidimensional scaling