Geometry of low nonnegative rank matrix completion
Kaie Kubjas, Lilja Metsälampi
TL;DR
This work systematically analyzes the geometry of completing partial nonnegative matrices to matrices of nonnegative rank at most $r$, with a focus on $r\in\{1,2,3\}$. It introduces a geometric framework based on nested polytopes and the elimination-ideal viewpoint to derive necessary conditions for rank-$r$ completions, and connects these ideas to algebraic matroids. The paper proves a tight equivalence for $r=1$ (nonnegative rank-1 completion ⇔ rank-1 completion) and a $3\times3$-pattern characterization for $r=2$, showing that if a partial matrix has a nonnegative rank-2 completion it also has a nonnegative rank-2 completion whenever a rank-2 completion exists, while providing counterexamples for $r=3$. It also analyzes one-entry missing scenarios via the projection's singular locus, offering a pathway to understand when rank-$r$ completions are possible and when they may be unique. Overall, the results deepen the understanding of nonnegative matrix completion and lay groundwork for further geometric and algorithmic investigations.
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).