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