Cramer-Rao lower bounds for low-rank decomposition of multidimensional arrays

Unlike low-rank matrix decomposition, which is generically nonunique for ra nk greater than one, low-rank three- and higher dimensional array decomposi tion is unique, provided that the array rank is lower than a certain bound, and the correct number of components (equal to array rank) is sought in th e decomposition. Parallel factor (PARAFAC) analysis is a common name for lo w-rank decomposition of higher dimensional arrays. This paper develops Cram er-Rao Bound (CRB) results for low-rank decomposition of three- and four-di mensional (3-D and 4-D) arrays, illustrates the behavior of the resulting b ounds, and compares alternating least squares algorithms that are commonly used to compute such decompositions with the respective CRBs. Simple-to-che ck necessary conditions for a unique low-rank decomposition are also provid ed.