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

Unlike low-rank matrix decomposition, which is generically nonunique for rank greater than one, low-rank three-and higher dimensional array decomposition 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 the decomposition. Parallel factor (PARAFAC) analysis is a common name for low-rank decomposition of higher dimensional arrays. This paper develops Cramer-Rao bound (CRB) results for low-rank decomposition of three- and four-dimensional (3-D and 4-D) arrays, illustrates the behavior of the resulting bounds, and compares alternating least squares algorithms that are commonly used to compute such decompositions with the respective CRBs. Simple-to-check necessary conditions for a unique low-rank decomposition are also provided.

I. Introduction

FOR matrices [two-dimensional (2-D) or two-way arrays], the low-rank property in itself is not enough to guarantee a unique data model, and one has to resort to additional problem-specific structural properties to obtain a unique parameterization. Examples include orthogonality (as in singular value decomposition), Vandermonde, Toeplitz, or finite-alphabet constraints. Notwithstanding the lack of inherent uniqueness, low-rank matrix decomposition plays a key role in modern signal processing.