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On the geometric interpretation of the nonnegative rank

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  • GILLIS, Nicolas

    ()
    (CORE, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium)

  • GLINEUR, François

    ()
    (CORE, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium)

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.

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Bibliographic Info

Paper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 2010051.

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Date of creation: 01 Oct 2010
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Handle: RePEc:cor:louvco:2010051

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Keywords: nonnegative rank; restricted nonnegative rank; nested polytopes; computational complexity; computational geometry; extended formulations; linear Euclidean distance matrices.;

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  1. Belleflamme,Paul & Peitz,Martin, 2010. "Industrial Organization," Cambridge Books, Cambridge University Press, number 9780521681599, April.
  2. GILLIS, Nicolas & GLINEUR, François, 2008. "Nonnegative factorization and the maximum edge biclique problem," CORE Discussion Papers 2008064, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. Winfried Pohlmeier & Luc Bauwens & David Veredas, 2007. "High frequency financial econometrics. Recent developments," ULB Institutional Repository 2013/136223, ULB -- Universite Libre de Bruxelles.
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