18 may 2016 -- 14:30
Aula Tricerri, DiMaI
Abstract.
The tensor rank decomposition problem consists of recovering the unique parameters of the decomposition from a robustly identifiable low-rank tensor. These parameters are subsequently analyzed and interpreted in many applications. As tensors are often perturbed by measurement errors in practice, one must investigate insofar 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 magnitude of the perturbation to the tensor. This paper introduces such a condition number for the tensor rank decomposition problem. It admits a closed expression as the inverse of a particular singular value of Terracini's matrix(a matrix representing the tangent space to the semi-algebraic set of tensors of fixed rank). A practical algorithm for computing the condition number is presented. The latter's elementary properties such as scaling and orthogonal invariance are established. The condition number of rank-1 tensors of order d equals $d^{1/2}$; they are always well-conditioned. The class of weak 3-orthogonal tensors, which includes orthogonally decomposable tensors, contains both well-conditioned and ill-conditioned problems. The numerical experiments confirm that the condition number yields a good upper bound on the magnitude of the change of the parameters when the tensor is perturbed. They also suggest that the condition number may be inversely related to the distance to ill-posed tensor rank decomposition problems, where the ill-posedness arises either from the nonclosedness of the set of tensors of fixed rank or from the existence of infinitely many decompositions.