Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

Information-geometric equivalence of transportation polytopes

  • Information Theory
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

This paper deals with transportation polytopes in the probability simplex (i.e., sets of categorical bivariate probability distributions with prescribed marginals). Information projections between such polytopes are studied, and a sufficient condition is described under which these mappings are homeomorphisms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Brualdi, R.A., Combinatorial Matrix Classes, Cambridge: Cambridge Univ. Press, 2006.

    Book  Google Scholar 

  2. Rüschendorf, L., Schweizer, B., and Taylor, M.D., Distributions with Fixed Marginals and Related Topics (Proc. AMS-IMS-SIAM Joint Summer Research Conf. on Distributions with Fixed Marginals, Doubly Stochastic Measures and Markov Operators, Seattle, WA, USA, Aug. 1–5, 1993), Hayward, CA, USA: Inst. Math. Statistics, 1996.

    Google Scholar 

  3. Csiszár, I. and Shields, P., Information Theory and Statistics: A Tutorial, Found. Trends Commun. Inf. Theory, 2004, vol. 1, no. 4, pp. 417–528.

    Article  Google Scholar 

  4. Polyanskiy, Y., Hypothesis Testing via a Comparator, in Proc. 2012 IEEE Int. Sympos. on Information Theory (ISIT’2012), Cambridge, MA, USA, July 1–6, 2012, pp. 2206–2210.

  5. Kovačević, M., Stanojević, I., and Šenk, V., On the Entropy of Couplings, Inform. Comput., 2015, vol. 242, pp. 369–382.

    Article  Google Scholar 

  6. Csiszár, I. and Körner, J., Information Theory: Coding Theorems for Discrete Memoryless Systems, New York: Academic; Budapest: Akad. Kiadó, 1981.

    Google Scholar 

  7. Chentsov, N.N., Nonsymmetrical Distance between Probability Distributions, Entropy and the Theorem of Pythagoras, Mat. Zametki, 1968, vol. 4, no. 3, pp. 323–332 [Math. Notes (Engl. Transl.), 1968, vol. 4, no. 3, pp. 686–691].

    MathSciNet  Google Scholar 

  8. Chentsov, N.N., Statisticheskie reshayushchie pravila i optimal’nye vyvody, Moscow: Nauka, 1972. Translated under the title Statistical Decision Rules and Optimal Inference, Providence, RI: Amer. Math. Soc., 1982.

    Google Scholar 

  9. Csiszár, I., I-Divergence Geometry of Probability Distributions and Minimization Problems, Ann. Probab., 1975, vol. 3, no. 1, pp. 146–158.

    Article  Google Scholar 

  10. Topsøe, F., Information Theoretical Optimization Techniques, Kybernetika, 1979, vol. 15, no. 1, pp. 8–27.

    MathSciNet  Google Scholar 

  11. Csiszár, I. and Matúš, F., Information Projections Revisited, IEEE Trans. Inform. Theory, 2003, vol. 49, no. 6, pp. 1474–1490.

    Article  MathSciNet  Google Scholar 

  12. Klee, V. and Witzgall, C., Facets and Vertices of Transportation Polytopes, Mathematics of the Decision Sciences, Part I (Seminar, Stanford, CA, 1967), Dantzig, G.B. and Veinott, A.F., Jr., Eds., Providence, RI: Amer. Math. Soc., 1968, pp. 257–282.

    Google Scholar 

  13. De Loera, J.A. and Kim, E.D., Combinatorics and Geometry of Transportation Polytopes: An Update, Discrete Geometry and Algebraic Combinatorics, Barg, A. and Musin O.R., Eds., Providence, RI: Amer. Math. Soc., 2014, pp. 37–76.

    Google Scholar 

  14. Gietl, C. and Reffel, F.P., Continuity of f-Projections on Discrete Spaces, Geometric Science of Information (Proc. 1st Int. Conf. (GSI’2013), Paris, France, Aug. 28–30, 2013), Nielsen, F. and Barbaresco, F., Eds., Lect. Notes Comp. Sci., vol. 8085, New York: Springer, 2013, pp. 519–524.

    Google Scholar 

  15. Bredon, G.E., Topology and Geometry, New York: Springer, 1993.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Kovačević.

Additional information

Original Russian Text © M. Kovačević, I. Stanojević, V. Šenk, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 2, pp. 20–26.

Supported by the Ministry of Education, Science, and Technological Development of the Republic of Serbia, grant nos. TR32040 and III44003.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kovačević, M., Stanojević, I. & Šenk, V. Information-geometric equivalence of transportation polytopes. Probl Inf Transm 51, 103–109 (2015). https://doi.org/10.1134/S0032946015020027

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946015020027

Keywords