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 hypersurf...

• Type: Journal
• Year: 2016
• Where: TIT
• Authors: Yang Qi, Pierre Comon, Lek-Heng Lim
• Keywords: Education, TIT 2016