A multilinear singular value decomposition. (English) Zbl 0962.15005

De Lathauwer, Lieven; De Moor, Bart; Vandewalle, Joos

SIAM J. Matrix Anal. Appl. 21, No. 4, 1253-1278 (2000).

This paper investigates higher order singular value decompositions (HOSVD) as multilinear generalization of singular value decomposition (SVD), in terms of higher order tensors exhibiting symmetries as motivated by applications in higher order statistics. This includes matrix representation of higher order tensors, their rank properties, generalization of scalar product, orthogonality, and Frobenius norm, and multiplication by a matrix. A HOSVD for real or complex \(N\)th order tensors is always possible and reduces to SVD when applied to a matrix. In psychometry it is called the Tucker model. The matrix of singular vectors can be computed from the sets of \(n\)-mode vectors in the same way as in the second order case.

Reviewer: E.Kreyszig (Ottawa)

MSC:

15A18	Eigenvalues, singular values, and eigenvectors
15A69	Multilinear algebra, tensor calculus
15A63	Quadratic and bilinear forms, inner products
15A60	Norms of matrices, numerical range, applications of functional analysis to matrix theory
62G30	Order statistics; empirical distribution functions
92B15	General biostatistics
62P15	Applications of statistics to psychology

Keywords:

multilinear algebra; singular value decomposition; higher-order tensor; higher order statistics; matrix representation; rank; scalar product; orthogonality; Frobenius norm; psychometry; Tucker model; singular vectors