The average condition number of most tensor rank decomposition problems is infinite

Tensor rank decomposition is the problem of computing a set of rank-1 tensors whose sum is a given tensor. We are interested in quantifying the sensitivity of real rank-1 summands when moving the tensor infinitesimally on the semialgebraic set of tensors of bounded real rank. For this purpose, the standard approach in numerical analysis consists of computing the condition number of this problem. If the condition number is infinite, then the problem is said to be ill-posed. In this talk, we present the condition number of tensor rank decomposition. For most ranks, we compute its average value over the semialgebraic set of real tensors of bounded rank, relative to a natural choice of probability distribution. The results show that the condition number blows up too fast in a neighborhood of ill-posed problems to result in a finite average value.