Coupled Canonical Polyadic Decompositions and (Coupled) Decompositions in Multilinear Rank- (Lᵣ, n, Lᵣ, n, 1) Terms---Part I: Uniqueness

Coupled tensor decompositions are becoming increasingly important in signal processing and data analysis. However, the uniqueness properties of coupled tensor decompositions have not yet been studied. In this paper, we first provide new uniqueness conditions for one factor matrix of the coupled canonical polyadic decomposition (CPD) of third-order tensors. Then, we present necessary and sufficient overall uniqueness conditions for the coupled CPD of third-order tensors. The results demonstrate that improved uniqueness conditions can indeed be obtained by taking into account the coupling between several tensor decompositions. We extend the results to higher-order tensors and explain that the higher-order structure can further improve the uniqueness results. We discuss the special case of coupled matrix-tensor factorizations. We also present a new variant of the coupled CPD model called the coupled block term decomposition (BTD). On one hand, the coupled BTD can be seen as a variant of coupled CPD for the case where the common factor contains collinear columns. On the other hand, it can also be seen as an extension of the decomposition into multilinear rank- (Lₑ, Lₑ, 1) terms to coupled factorizations.