The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite

Abstract The tensor rank decomposition, or canonical polyadic is the decomposition of a into sum rank-1 tensors. condition number measures sensitivity summands with respect to structured perturbations. Those are perturbations preserving that decomposed. On other hand, angular up scaling. We show for random rank-2 tensors expected value infinite wide range choices density. Under mild additional assumption, we same true most higher ranks $$r\ge 3$$ r ≥ 3 as well. In fact, dimensions tend infinity, asymptotically all covered by our analysis. contrary, have finite number. Based on numerical experiments, conjecture this could also be ranks. Our results underline high computational complexity computing decompositions. discuss consequences algorithm design and testing algorithms