UNIQUENESS CONDITIONS FOR CONSTRAINEDTHREE-WAY FACTOR DECOMPOSITIONS WITH LINEARLYDEPENDENT LOADINGS

摘要

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