Semialgebraic Geometry of Nonnegative Tensor Rank

TL;DR

This paper explores the semialgebraic structure of nonnegative tensor rank, establishing properties like typical ranks, uniqueness of decompositions, and the validity of the direct sum conjecture for nonnegative tensors.

Contribution

It provides a comprehensive analysis of nonnegative tensor rank, including determination of typical ranks, proof of the direct sum conjecture, and conditions for unique decompositions.

Findings

All nonnegative typical ranks for cubical tensors are determined.

The direct sum conjecture holds for nonnegative tensor rank.

Unique nonnegative rank-$r$ decompositions exist under certain conditions.

Abstract

We study the semialgebraic structure of $D_r$, the set of nonnegative tensors of nonnegative rank not more than $r$, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for cubical nonnegative tensors and show that the direct sum conjecture is true for nonnegative tensor rank. We show that nonnegative, real, and complex ranks are all equal for a general nonnegative tensor of nonnegative rank strictly less than the complex generic rank. In addition, such nonnegative tensors always have unique nonnegative rank-$r$ decompositions if the real tensor space is $r$-identifiable. We determine conditions under which a best nonnegative rank-$r$ approximation has a unique nonnegative rank-$r$ decomposition: for $r \le 3$, this is always the case; for general $r$, this is the case when the best nonnegative rank-$r$ approximation does not lie on the boundary of $D_r$. Many of our general identifiability results also apply to real tensors and real symmetric tensors.