Improved Uniqueness Conditions for Canonical Tensor Decompositions with Linearly Dependent Loadings

In this paper, we derive improved uniqueness conditions for a constrained version of the canonical order-3 tensor decomposition, also known as Candecomp/Parafac (CP). 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 decompositions. We provide both uniqueness conditions and partial uniqueness conditions for PARALIND, and show that these are improved and more precise variants of existing conditions. Our results are illustrated by means of examples.