Maximal and typical nonnegative ranks of nonnegative tensors

TL;DR

This paper characterizes the typical and maximal nonnegative ranks of tensors with given dimensions, showing that the maximal nonnegative rank equals the product of the first d-1 dimensions and relates typical ranks to the complex generic rank.

Contribution

It provides a complete characterization of the typical nonnegative ranks and establishes that the maximal nonnegative rank equals the product of the first d-1 dimensions.

Findings

Maximal nonnegative rank equals the product of the first d-1 dimensions.

Typical nonnegative ranks are bounded above by this maximum and below by the complex generic rank.

The results unify the understanding of nonnegative tensor ranks across different formats.

Abstract

Let $N_1, \ldots, N_d$ be positive integers with $N_1\leq\cdots\leq N_d$. Set $N=N_1\cdots N_{d-1}$. We show in this paper that an integer $r$ is a typical nonnegative rank of nonnegative tensors of format $N_1\times\cdots\times N_d$ if and only if $r\leq N$ and $r$ is greater than or equals to the generic rank of tensors over $\mathbb{C}$ of format $N_1\times\cdots\times N_d$. We also show that the maximal nonnegative rank of nonnegative tensors of format $N_1\times\cdots\times N_d$ is $N$.