Geometry of low nonnegative rank matrix completion

TL;DR

This paper investigates the geometric and combinatorial properties of low nonnegative rank matrix completion, providing characterizations and examples mainly for ranks 1 to 3, and extending geometric insights into nested polytopes.

Contribution

It offers new geometric characterizations for nonnegative rank-$r$ completions and identifies conditions under which rank and nonnegative rank completions coincide for low ranks.

Findings

Rank-1 completion characterized by rank-1 completion.

For 3x3 matrices, patterns for rank-2 and nonnegative rank-2 equivalence.

Examples showing differences between rank and nonnegative rank for rank 3.

Abstract

We study completion of partial matrices with nonnegative entries to matrices of nonnegative rank at most $r$ for some $r \in \mathbb{N}$. Most of our results are for $r \leq 3$. We show that a partial matrix with nonnegative entries has a nonnegative rank-1 completion if and only if it has a rank-1 completion. This is not true in general when $r \geq 2$. For $3 \times 3$ matrices, we characterize all the patterns of observed entries when having a rank-2 completion is equivalent to having a nonnegative rank-2 completion. If a partial matrix with nonnegative entries has a rank-$r$ completion that is nonnegative, where $r \in \{1,2\}$, then it has a nonnegative rank-$r$ completion. We will demonstrate examples for $r=3$ where this is not true. We do this by introducing a geometric characterization for nonnegative rank-$r$ completion employing families of nested polytopes which generalizes the geometric characterization for nonnegative rank introduced by Cohen and Rothblum (1993).