Uniqueness conditions for constrained three-way factor decompositions with linearly dependent loadings

In this paper, we derive uniqueness conditions for a constrained version of the parallel factor (Parafac) decomposition, also known as canonical decomposition (Candecomp). Candecomp/Parafac (CP) decomposes a three-way array into a prespecified number of outer product arrays. The constraint is that some vectors forming the outer product arrays are linearly dependent according to a prespecified pattern. This is known as the PARALIND family of models. An important subclass is where some vectors forming the outer product arrays are repeated according to a prespecified pattern. These are known as CONFAC decompositions. We discuss the relation between PARALIND, CONFAC, and the three-way decompositions CP, Tucker3, and the decomposition in block terms. We provide both essential uniqueness conditions and partial uniqueness conditions for PARALIND and CONFAC and discuss the relation with uniqueness of constrained Tucker3 models and the block decomposition in rank-(L, L, 1) terms. Our results are demonstrated by means of examples.