Canonical polyadic decomposition of third-order tensors: reduction to generalized eigenvalue decomposition (Q2923363)

Canonical polyadic decomposition (CPD) of a third-order tensor is the decomposition in a minimal number of rank-1 tensors. The known algebraic algorithms for the computation of the CPD are limited to cases where at least one of the factor matrices has full column rank. A novel approach is presented in this paper which is applicable in cases where none of the factor matrices has full column rank. The procedure is proved for real and exact problems. Whereas the complex numbers could be easily incorporated, the extension for inexact (noisy) problems is only discussed in brief.NEWLINENEWLINETwo algorithms are proposed. Both of them are algebraic in the sense that they rely only on the standard linear algebra operations. Both reduce the problem to the computation of a generalized eigenvalue decomposition of a matrix pencil. They differ by the computational complexity for particular problems.