Pencil-Based Algorithms For Tensor Rank Decomposition Are Not Stable

Abstract

We prove the existence of an open set of n1 ×n2 ×n3 tensors of rank r for which popular and e?cient algorithms for computing tensor rank decompositions based on a reduction to a linear matrix pencil, typically followed by a generalized eigendecomposition, are arbitrarily numerically forward unstable. Our analysis shows that this problem is caused by the fact that the condition number of tensor rank decomposition can be much larger for n1 ×n2 ×2 tensors than for the n1 ×n2 ×n3 input tensor. Moreover, we present a lower bound for the limiting distribution of the condition number of random tensor rank decompositions of third-order tensors. The numerical experiments illustrate that for random tensor rank decompositions one should anticipate a loss of precision of a few digits.