Tensor Networks for Latent Variable Analysis: Higher Order Canonical Polyadic Decomposition

The canonical polyadic decomposition (CPD) is a convenient and intuitive tool for tensor factorization; however, for higher order tensors, it often exhibits high computational cost and permutation of tensor entries, and these undesirable effects grow exponentially with the tensor order. Prior compression of tensor in-hand can reduce the computational cost of CPD, but this is only applicable when the rank
$R$
of the decomposition does not exceed the tensor dimensions. To resolve these issues, we present a novel method for CPD of higher order tensors, which rests upon a simple tensor network of representative inter-connected core tensors of orders not higher than 3. For rigor, we develop an exact conversion scheme from the core tensors to the factor matrices in CPD and an iterative algorithm of low complexity to estimate these factor matrices for the inexact case. Comprehensive simulations over a variety of scenarios support the proposed approach.