On the Geometric Interpretation of the Nonnegative Rank

By Nicolas Gillis, Franccois Glineur. 2010

92 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/1befc8e1fa49e767a213ed7de056128d117f81f0

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In Computational Mechanics

In Computational Theory and Mathematics

Nicolas Gillis

In Computer Vision and Pattern Recognition

Citations in the corpus

Detecting Memorization in ReLU Networks (2018, background): “For the support of a non-negative matrix, a well-known upper-bound is: rc(supp(A)) ≤ rank+(A), (2) where rank+(A) is the non-negative rank of A (Gillis & Glineur, 2012) that is defined as the smallest number k such that there exist non-negative matrices U+ ∈ Rn×k+ , V+ ∈ R k×q + such that A = U+V+.…”
The Why and How of Nonnegative Matrix Factorization (2014, background): “Computing the nonnegative rank is closely related to the problem of ﬁnding a polytope with minimum number of vertices nested between two given polytopes [53].”