Stability of CANDECOMP-PARAFAC tensor decomposition

In this paper, stability of the CANDECOMP-PARAFAC (CP) tensor decomposition is addressed. It is done by deriving the Cramer-Rao lower bound (CRLB) on variance of an unbiased estimate of the tensor parameters, i.e. elements of its factor matrices, from its noisy observation (the tensor plus a random Gaussian i.i.d. tensor). The existence of the bound reveals necessary conditions for essential uniqueness of the CP decomposition, moreover, for identifiability of each column of each factor matrix separately. Analytical closed-form expressions of the bound are derived for 3 way tensors of rank 1 and 2. As a byproduct, a novel computationally efficient expression for the inverse of the approximate Hessian matrix is derived.