Condition numbers for the tensor rank decomposition

The tensor rank decomposition problem consists of recovering the parameters of the model from an identifiable low-rank tensor. These parameters are analyzed and interpreted in many applications. As tensors are often perturbed by measurement errors in practice, one must investigate to what extent the unique parameters change in order to preserve the validity of the analysis. The magnitude of this change can be bounded asymptotically by the product of the condition number and the norm of the perturbation to the tensor. This paper introduces a condition number that admits a closed expression as the inverse of a particular singular value of Terracini's matrix, which represents the tangent space to the set of tensors of fixed rank. A practical algorithm for computing this condition number is presented. The latter's elementary properties such as scaling and orthogonal invariance are established. Rank-1 tensors are always well-conditioned. The class of weak 3-orthogonal tensors, which includes orthogonally decomposable tensors, contains both well-conditioned and ill-conditioned problems. Numerical experiments confirm that the condition number yields a good estimate of the magnitude of the change of the parameters when the tensor is perturbed.