[論文レビュー] Geometry of low nonnegative rank matrix completion

Kaie Kubjas, Lilja Metsälampi|arXiv (Cornell University)|Jan 12, 2026

Advanced Optimization Algorithms Research被引用数 0

ひとこと要約

The paper studies completing partial nonnegative matrices to matrices with nonnegative rank at most r, giving results for r ≤ 3, including a rank-1 completion equivalence with ordinary rank-1 completion and a geometric characterization via nested polytopes for nonnegative rank-r completeness.

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 $3 imes 3$ 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).

研究の動機と目的

• Motivate low nonnegative rank matrix completion in contexts with missing data (e.g., recommender systems, image processing).
• Characterize when partial nonnegative matrices admit nonnegative rank-1 and rank-2 completions.
• Develop and apply a geometric framework via nested polytopes to understand nonnegative rank-r completions (r ≤ 3).
• Identify limitations for r ≥ 3 through explicit examples and discuss implications for uniqueness and algorithms.

提案手法

• Use geometric description of nonnegative rank via nested cones and polytopes (P ⊆ Δ ⊆ Q) to analyze completions.
• Prove that a partial nonnegative matrix has a nonnegative rank-1 (r=1) completion iff it has a rank-1 completion.
• Characterize 3×3 patterns for r=2 when rank-2 completion implies nonnegative rank-2 completion.
• Show that if a partial nonnegative matrix has a rank-r nonnegative completion with r∈{1,2}, then it has a nonnegative rank-r completion.
• Introduce a geometric characterization for nonnegative rank-r completion using families of nested polytopes (generalizing Cohen–Rothblum and Vavasis framework).
• Provide examples to illustrate when the r=3 case fails (one and two missing entries).

実験結果

リサーチクエスチョン

• RQ1 When does a partial nonnegative matrix admit a nonnegative rank-r completion?
• RQ2 Does every rank-r completion that is nonnegative automatically qualify as a nonnegative rank-r completion for r≤2?
• RQ3 For 3×3 matrices, which observed-entry patterns guarantee that a rank-2 completion implies a nonnegative rank-2 completion?
• RQ4 Are rank-1 and rank-2 nonnegative completions always guaranteed to exist when a (nonnegative) rank-r completion exists for r≤2?
• RQ5 How does the geometry (nested polytopes) characterize nonnegative rank-r completions, and what does this imply about r≥3?
• RQ6 What are explicit counterexamples for r=3 demonstrating limitations of equivalences observed for r≤2?

主な発見

• A partial matrix with nonnegative entries has a nonnegative rank-1 completion iff it has a rank-1 completion.
• For 3×3 matrices, there is a characterization of observed-entry patterns where a rank-2 completion is equivalent to a nonnegative rank-2 completion.
• If a partial matrix with nonnegative entries has a rank-r completion that is nonnegative with r∈{1,2}, then it has a nonnegative rank-r completion.
• The equivalence between rank-2 completion and nonnegative rank-2 completion can fail for r=3, demonstrated by explicit examples.
• A geometric characterization of nonnegative rank-r completion via families of nested polytopes generalizes the Cohen–Rothblum/Vavasis approach for nonnegative rank, enabling a broader analysis of completions.
• Examples show that for r=3 there can be partial matrices with a rank-3 completion that is nonnegative but without a nonnegative rank-3 completion (and similarly with two missing entries).