Uniqueness of Nonnegative Tensor Approximations

We show that for a nonnegative tensor, a best nonnegative rank-r approximation is almost always unique, its best rank-one approximation may always be chosen to be a best nonnegative rank-one approximation, and that the set of nonnegative tensors with non-unique best rank-one approximations form an algebraic hypersurface. We show that the last part holds true more generally for real tensors and thereby determine a polynomial equation so that a real or nonnegative tensor which does not satisfy this equation is guaranteed to have a unique best rank-one approximation. We also establish an analogue for real or nonnegative symmetric tensors. In addition, we prove a singular vector variant of the Perron--Frobenius Theorem for positive tensors and apply it to show that a best nonnegative rank-r approximation of a positive tensor can never be obtained by deflation. As an aside, we verify that the Euclidean distance (ED) discriminants of the Segre variety and the Veronese variety are hypersurfaces and give defining equations of these ED discriminants

The average number of critical rank-one approximations to a tensor

Motivated by the many potential applications of low-rank multi-way tensor approximations, we set out to count the rank-one tensors that are critical points of the distance function to a general tensor v. As this count depends on v, we average over v drawn from a Gaussian distribution, and find formulas that relates this average to problems in random matrix theory.Comment: Several minor edit

On the average condition number of tensor rank decompositions

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