Cramer-Rao-Induced Bounds for CANDECOMP/PARAFAC tensor decomposition

This paper presents a Cramer-Rao lower bound (CRLB) on the variance of unbiased estimates of factor matrices in Canonical Polyadic (CP) or CANDECOMP/PARAFAC (CP) decompositions of a tensor from noisy observations, (i.e.,the tensor plus a random Gaussian i.i.d. tensor). A novel expression is derived for a bound on the mean square angular error of factors along a selected dimension of a tensor of an arbitrary dimension. Insightful expressions are derived for tensors of rank 1 and rank 2. The existence of the bound reveals necessary conditions for essential uniqueness of the CP decomposition and, moreover, for identifiability of each column of each factor matrix separately. The results can be used for checking stability of a given decomposition of a tensor, and for evaluating performance of certain approximate CP decomposition methods based on reshaping the tensor.