We compute the expected value of powers of the geometric condition number of
random tensor rank decompositions. It is shown in particular that the expected
value of the condition number of n1βΓn2βΓ2 tensors with a random
rank-r decomposition, given by factor matrices with independent and
identically distributed standard normal entries, is infinite. This entails that
it is expected and probable that such a rank-r decomposition is sensitive to
perturbations of the tensor. Moreover, it provides concrete further evidence
that tensor decomposition can be a challenging problem, also from the numerical
point of view. On the other hand, we provide strong theoretical and empirical
evidence that tensors of size n1βΒ ΓΒ n2βΒ ΓΒ n3β with all n1β,n2β,n3ββ₯3 have a finite average condition number. This suggests there exists a gap
in the expected sensitivity of tensors between those of format n1βΓn2βΓ2 and other order-3 tensors. For establishing these results, we show
that a natural weighted distance from a tensor rank decomposition to the locus
of ill-posed decompositions with an infinite geometric condition number is
bounded from below by the inverse of this condition number. That is, we prove
one inequality towards a so-called condition number theorem for the tensor rank
decomposition