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 \(r\ge 3\) 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.

张量秩分解或规范多元分解是将张量分解为秩为 1 的张量之和。张量秩分解的条件数测量秩 1 和对结构化扰动的敏感性。这些是保持分解张量等级的扰动。另一方面，角度条件数测量 rank-1 和的扰动直到缩放。我们展示了随机 rank-2 张量，对于各种密度选择，条件数的期望值是无限的。在一个温和的附加假设下，我们表明对于大多数较高等级\(r\ge 3\)也是。事实上，由于张量的维数趋于无穷大，我们的分析逐渐涵盖了所有秩。相反，我们证明了 rank-2 张量具有有限的预期角条件数。基于数值实验，我们推测这也可能适用于更高的等级。我们的结果强调了计算张量秩分解的高计算复杂性。我们讨论了我们的结果对算法设计和测试计算张量秩分解的算法的影响。