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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The authors derive the Cramer-Rao lower bound (CRLB) for complex signals with constant amplitude and polynomial phase, measured in additive Gaussian white noise. The exact bound requires numerical inversion of an ill-conditioned matrix, while its O(N sup -1 ) approximation is free of matrix inversion. The approximation is tested for several typical parameter values and is found to be excellent in most cases. The formulas derived are of practical value in several radar applications, such as electronic intelligence systems (ELINT) for special pulse-compression radars, and motion estimation from Doppler measurements. Consequently, it is of interest to analyze the best possible performance of potential estimators of the phase coefficients, as a function of signal parameters, the signal-to-noise ratio, the sampling rate, and the number of measurements. This analysis is carried out. > 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. Many idealized problems in signal processing, machine learning, and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing inspiration from the matrix case, the successive rank-one approximation (SROA) scheme has been proposed and shown to yield this tensor decomposition exactly, and a plethora of numerical methods have thus been developed for the tensor rank-one approximation problem. In practice, however, the inevitable errors (say) from estimation, computation, and modeling necessitate that the input tensor can only be assumed to be a nearly SOD tensor---i.e., a symmetric tensor slightly perturbed from the underlying SOD tensor. This paper shows that even in the presence of perturbation, SROA can still robustly recover the symmetric canonical decomposition of the underlying tensor. It is shown that when the perturbation error is small enough, the approximation errors do not accumulate with ... Abstract We derive expressions for the Cramer-Rao Bound (CRB) for the estimates of the parameters of a deterministic signal observed in additive and multiplicative noise which may be i.i.d. non-Gaussian or colored Gaussian. General expressions for the CRB are derived, and are then applied to the specific case of polynomial phase signals for which closed-form expressions are obtained under certain assumptions. We also develop bounds on the CRB; these bounds are tight at low SNR.