On the average condition number of tensor rank decompositions
For researchers in numerical linear algebra and tensor methods, it quantifies the expected sensitivity of tensor decompositions, showing a fundamental difference between 2-slice and higher-order tensors.
The paper computes the expected condition number of random tensor rank decompositions, finding it is infinite for 2-slice tensors (n1×n2×2) but finite for tensors with all dimensions ≥3, indicating a sensitivity gap. This provides theoretical evidence that tensor decomposition can be numerically challenging.
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 $n_1\times n_2 \times 2$ 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 $n_1~\times~n_2~\times~n_3$ with all $n_1,n_2,n_3 \ge 3$ have a finite average condition number. This suggests there exists a gap in the expected sensitivity of tensors between those of format $n_1\times n_2 \times 2$ 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.