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The average condition number of most tensor rank decomposition problems is infinite
Carlos Beltrán, Paul Breiding and Nick Vannieuwenhoven
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 with Gaussian density that the expected value of the condition number is infinite. Under some 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 Gaussian tensors have finite expected angular condition number.
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 that compute the CPD. Finally, we supply numerical experiments.