On the Geometric Interpretation of the Nonnegative Rank

TL;DR

This paper explores the geometric aspects of the nonnegative rank of matrices, introduces the restricted nonnegative rank, and uses these insights to improve bounds and counterexamples in matrix theory and optimization.

Contribution

It introduces the restricted nonnegative rank, characterizes its complexity, and applies these findings to provide new bounds and counterexamples in matrix analysis.

Findings

Computing the restricted nonnegative rank is equivalent to a polyhedral combinatorics problem.

New lower bounds for nonnegative rank are derived from geometric interpretation.

Counterexamples show nonnegative rank of Euclidean distance matrices can be less than their dimension.

Abstract

The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining, graph theory and computational geometry. In particular, it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in polyhedral combinatorics, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasly and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square.