IDEAS home Printed from
   My bibliography  Save this paper

On the geometric interpretation of the nonnegative rank


  • 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)


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.

Suggested Citation

  • GILLIS, Nicolas & GLINEUR, François, 2010. "On the geometric interpretation of the nonnegative rank," CORE Discussion Papers 2010051, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvco:2010051

    Download full text from publisher

    File URL:
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    1. 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).
    2. Rabah Amir, 2005. "Supermodularity and Complementarity in Economics: An Elementary Survey," Southern Economic Journal, Southern Economic Association, vol. 71(3), pages 636-660, January.
    3. Winfried Pohlmeier & Luc Bauwens & David Veredas, 2007. "High frequency financial econometrics. Recent developments," ULB Institutional Repository 2013/136223, ULB -- Universite Libre de Bruxelles.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. repec:spr:jglopt:v:68:y:2017:i:2:d:10.1007_s10898-016-0466-9 is not listed on IDEAS
    2. Arnaud Vandaele & Nicolas Gillis & François Glineur & Daniel Tuyttens, 2016. "Heuristics for exact nonnegative matrix factorization," Journal of Global Optimization, Springer, vol. 65(2), pages 369-400, June.

    More about this item


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

    JEL classification:

    • A23 - General Economics and Teaching - - Economic Education and Teaching of Economics - - - Graduate
    • B11 - Schools of Economic Thought and Methodology - - History of Economic Thought through 1925 - - - Preclassical (Ancient, Medieval, Mercantilist, Physiocratic)
    • D99 - Microeconomics - - Micro-Based Behavioral Economics - - - Other
    • F30 - International Economics - - International Finance - - - General

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cor:louvco:2010051. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Alain GILLIS). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.