On the Uniqueness of the Canonical Polyadic Decomposition of Third-Order Tensors - Part II: Uniqueness of the Overall Decomposition

Canonical polyadic (also known as Candecomp/Parafac) decomposition (CPD) of a higher-order tensor is decomposition into 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. We consider both deterministic and generic uniqueness. We also discuss uniqueness of INDSCAL and other constrained polyadic decompositions.

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