First-order perturbation analysis of the best rank-(R1,R2,R3) approximation in multilinear algebra

In this paper we perform a first-order perturbation analysis of the least squares approximation of a given higher-order tensor by a tensor having prespecified n-mode ranks. This work generalizes the classical first-order perturbation analysis of the matrix singular value decomposition. We will show that there are important differences between the matrix and the higher-order tensor case. We subsequently address (1) the best rank-1 approximation of supersymmetric tensors, (2) the best rank-(R 1 , R 2 , R 3 ) approximation of arbitrary tensors and (3) the best rank-(R 1 , R 2 , R 3 ) approximation of arbitrary tensors.