On the geometric interpretation of the nonnegative rank
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.
|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
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.:
- AMIR, Rabah, 2003.
"Supermodularity and complementarity in economics: an elementary survey,"
CORE Discussion Papers
2003104, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Rabah Amir, 2005. "Supermodularity and Complementarity in Economics: An Elementary Survey," Southern Economic Journal, Southern Economic Association, vol. 71(3), pages 636-660, January.
- AMIR, Rabah, . "Supermodularity and complementarity in economics: an elementary survey," CORE Discussion Papers RP 1823, 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.
- 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).
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)
If references are entirely missing, you can add them using this form.