Uniqueness of Nonnegative Tensor Approximations
Provides foundational theoretical guarantees for uniqueness of nonnegative tensor approximations, relevant to tensor decomposition and optimization.
The paper proves that best nonnegative rank-r approximations of nonnegative tensors are almost always unique, and characterizes when uniqueness fails via an algebraic hypersurface. It also shows that deflation cannot yield best nonnegative rank-r approximations for positive tensors.
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.