Uniqueness of Nonnegative Tensor Approximations

TL;DR

This paper proves the almost always uniqueness of best nonnegative tensor approximations, characterizes when non-uniqueness occurs, and extends classical theorems to nonnegative and symmetric tensors, with implications for tensor approximation methods.

Contribution

It establishes the generic uniqueness of nonnegative tensor approximations, characterizes the algebraic structure of non-uniqueness, and extends Perron-Frobenius theory to positive tensors.

Findings

Best nonnegative tensor approximations are almost always unique.

Non-uniqueness of rank-one approximations forms an algebraic hypersurface.

A polynomial equation guarantees uniqueness of best rank-one approximations for real tensors.

Abstract

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.