The typical rank of tall three-way arrays

The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank of I x J x K arrays with K = 2 to a particular class of arrays with K greater than or equal to 2. It is shown that the typical rank is I when the array is tall in the sense that JK - J <I <JK. In addition, typical rank results are given for the case where I equals JK - J.