On the Uniqueness of the Canonical Polyadic Decomposition of third-order tensors --- Part II: Uniqueness of the overall decomposition

TL;DR

This paper extends previous work on the uniqueness of the Canonical Polyadic Decomposition (CPD) of third-order tensors, providing new conditions for overall uniqueness when factor matrices lack full column rank.

Contribution

It introduces new uniqueness conditions for CPD of tensors with non-full-rank factor matrices, involving Khatri-Rao products and Kruskal-type criteria.

Findings

Established conditions for overall CPD uniqueness without full column rank

Utilized Khatri-Rao products of compound matrices in proofs

Extended previous results to broader classes of tensors

Abstract

Canonical Polyadic (also known as Candecomp/Parafac) Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of rank-1 tensors. In Part I, we gave an overview of existing results concerning uniqueness and presented new, relaxed, conditions that guarantee uniqueness of one factor matrix. In Part II we use these results for establishing overall CPD uniqueness in cases where none of the factor matrices has full column rank. We obtain uniqueness conditions involving Khatri-Rao products of compound matrices and Kruskal-type conditions.