On the geometric interpretation of the nonnegative rank

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 and graph theory. In particular, it can be used to characterize the minimal size of any extended reformulation of a given polytope. 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 computational geometry, 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 Beasley 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.

An upper bound for nonnegative rank

“…Lin and Chu [11] claimed a positive answer for Problem 1.3, but their argument has been shown to contain a gap [8,9]. A negative answer for Problem 1.3 has been obtained in [8] for a special case of so-called Euclidean distance matrices.…”

“…An extension [5,8] of P is a polytope Q ⊂ R d such that P can be obtained from Q as an image under a linear projection from R d to R n . An extended formulation [8,10] of P is a description of Q by linear equations and linear inequalities (together with the projection). The size [8, 10] of the extended formulation is the number of facets of Q.…”

“…Nonnegative factorizations are being widely studied and used in data analysis, statistics, computational biology, clustering and numerous other applications [2]. There are still many open questions on nonnegative rank interesting for different applications, and a considerable part of them is related to providing the bounds on the nonnegative rank in terms of other matrix invariants [4,8,10].…”

“…In view of Proposition 1.1 and Theorem 1.2, one can ask a related question on whether there exists a convex n-gon with extension complexity equal to n, for every n. For n ≤ 5, Problem 1.3 has been solved in the positive in [8]. In [6] it was noted that a sufficiently irregular convex hexagon has full extension complexity, stating the positive answer for n = 6.…”

“…The result by Yannakakis points out the connection between extension complexity and nonnegative factorizations and can now be formulated as follows [8,10,12].…”