The tensor rank decomposition problem consists of recovering the unique set of parameters representing a robustly identifiable low-rank tensor when the coordinate representation of the tensor is presented as input. A condition number for this problem measuring the sensitivity of the parameters to an infinitesimal change to the tensor is introduced and analyzed. It is demonstrated that the absolute condition number coincides with the inverse of the least singular value of Terracini's matrix. Several basic properties of this condition number are investigated.
Perhaps the result closest in spirit to this work are the Cram 'er--Rao bounds (CRB) for the tensor rank decomposition problem that have been investigated in @cite_78 @cite_2 @cite_67 @cite_75 . This CRB measures the stability of a tensor rank decomposition in a statistical framework, wherein additive Gaussian noise is assumed to corrupt the factor matrices. The quantity of interest is the squared angular error between the true parameters and the estimated parameters @cite_75 . The Cram 'er--Rao lower bound (CRLB) for estimating the parameters @math of the tensor rank decomposition of @math is then defined @cite_75 as the inverse of the Fisher information matrix @math , where @math is as in thm_informal_theorem and @math is the variance of the Gaussian noise. However, this matrix is not invertible, so in @cite_75 it is suggested to alter the tensor rank decomposition problem by removing some of the parameters. This implies that the CRLB depends on the particular elimination of the variables that is chosen; hence, it is not an measure of stability of the tensor rank decomposition problem as defined in this paper.
@cite_67 @cite_75 @cite_78 @cite_2
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In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. 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. INDSCAL is a special case of the CANDECOMP-PARAFAC (CP) decomposition of three or more-way tensors, where two factor matrices are equal. This paper provides a stability analysis of INDSCAL that is done by deriving the Cramer-Rao lower bound (CRLB) on variance of an unbiased estimate of the tensor parameters from its noisy observation (the tensor plus an i.i.d. Gaussian random tensor). The existence of the bound reveals necessary conditions for the essential uniqueness of the INDSCAL decomposition. This is compared with previous results on CP. Next, analytical expressions for the inverse of the Hessian matrix, which is needed to compute the CRLB, are used in a damped Gaussian (Levenberg-Marquardt) algorithm, which gives a novel method for INDSCAL having a lower computational complexity.