On the geometric interpretation of the nonnegative rank
AbstractThe 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.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Bibliographic InfoPaper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 2010051.
Date of creation: 01 Oct 2010
Date of revision:
Contact details of provider:
Postal: Voie du Roman Pays 34, 1348 Louvain-la-Neuve (Belgium)
Fax: +32 10474304
Web page: http://www.uclouvain.be/core
More information through EDIRC
nonnegative rank; restricted nonnegative rank; nested polytopes; computational complexity; computational geometry; extended formulations; linear Euclidean distance matrices.;
Other versions of this item:
- GILLIS, Nicolas & GLINEUR, François, . "On the geometric interpretation of the nonnegative rank," CORE Discussion Papers RP -2439, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- NEP-ALL-2011-02-12 (All new papers)
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Belleflamme,Paul & Peitz,Martin, 2010. "Industrial Organization," Cambridge Books, Cambridge University Press, number 9780521681599, April.
- 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).
- GILLIS, Nicolas & GLINEUR, François, 2010. "Nonnegative factorization and the maximum edge biclique problem," CORE Discussion Papers 2010059, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Winfried Pohlmeier & Luc Bauwens & David Veredas, 2007. "High frequency financial econometrics. Recent developments," ULB Institutional Repository 2013/136223, ULB -- Universite Libre de Bruxelles.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Alain GILLIS).
If references are entirely missing, you can add them using this form.