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 the set of nonnegative tensors with nonunique best rank-one approximations forms 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 that 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.