Nonnegative rank depends on the field

TL;DR

This paper demonstrates that the nonnegative rank of a matrix can vary depending on the underlying field, showing that the choice of field affects matrix factorization properties.

Contribution

It provides a specific example illustrating that nonnegative rank is field-dependent, which was previously not well understood.

Findings

Nonnegative rank varies with the field of entries.

Existence of matrices with different nonnegative ranks over different fields.

Field dependence impacts matrix factorization theory.

Abstract

We present an example of a subfield $\mathcal{F}\subset\mathbb{R}$ and a matrix $A$ whose conventional and nonnegative ranks equal five, but the nonnegative rank with respect to $\mathcal{F}$ equals six. In other words, $A$ can be represented as a sum of five rank-one matrices with nonnegative real entries but not as a sum of five rank-one matrices with nonnegative entries in $\mathcal{F}$.