Information Geometry of Barycenter Map

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)


Using barycenter of the Busemann function we define a map, called the barycenter map from a space \(\mathcal{P}^{+}\) of probability measures on the ideal boundary ∂ X to an Hadamard manifold X. We show that the space \(\mathcal{P}^{+}\) carries a fibre space structure over X from a viewpoint of information geometry. Following the idea of [7, 9] and [8] we present moreover a theorem which states that under certain hypotheses of information geometry a homeomorphism Φ of ∂ X induces, via the push-forward for probability measures, an isometry of X whose ∂ X-extension coincides with Φ.


Probability Measure Poisson Kernel Ideal Boundary Fibre Space Information Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors express their congratulatory address to Professor Young Jin Suh being selected as one of distinguished present mathematicians in Korea and the first author also is grateful to the organizing committee of ICMRCS2014, especially to Professor Suh.


  1. 1.
    Amari, S., Nagaoka, H.: Methods of Information Geometry. Transactions of Mathematical Monographs, vol. 191. AMS, Oxford (2000)Google Scholar
  2. 2.
    Anker, J.-P., Damek, E., Yacoub, C.: Spherical analysis on harmonic AN groups. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23, 643–679 (1996)MATHMathSciNetGoogle Scholar
  3. 3.
    Arnaudon, M., Barbaresco, F., Yang, L.: Medians and Means in Riemannian Geometry: Existence, Uniqueness and Computation. In: Nielson, F., Bhatia, R. (eds.) Matrix Information Geometry, pp. 169–197. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Ballmann, W., Gromov, M., Schroeder, V.: Manifolds of Nonpositive Curvature. Progress in Mathematics, vol. 61. Birkhäuser, Boston (1985)Google Scholar
  5. 5.
    Barbaresco, F.: Information Geometry of Covariance Matrix: Cartan-Siegel Homogeneous Bounded Domains, Mostow/Berger Fibration and Fréchet Median. In: Nielson, F., Bhatia, R. (eds.) Matrix Information Geometry, pp. 199–255. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Berndt, J., Tricerri, F., Vanhecke, L.: Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces. Lecture Notes in Mathematics, vol. 1598. Springer, Heidelberg (1991)Google Scholar
  7. 7.
    Besson, G., Courtois, G., Gallot, S.: Entropies et Rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5, 731–799 (1995)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Besson, G., Courtois, G., Gallot, S.: Minimal entropy and Mostow’s rigidity theorems. Ergodic Theory Dyn. Sys. 16, 623–649 (1996)MATHMathSciNetGoogle Scholar
  9. 9.
    Douady, E., Earle, C.: Conformally natural extension of homeomorphisms of the circle. Acta Math. 157, 23–48 (1986)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Eberlein, P., O’Neill, B.: Visibility manifolds. Pac. J. Math. 46, 45–110 (1973)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Friedrich, T.: Die Fisher-Information and symplektische Strukturen. Math. Nachr. 153, 273–296 (1991)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Guivarc’h, Y., Ji, L., Taylor, J.C.: Compactifications of Symmetric Spaces. Progress in Mathematics, vol. 156. Birkhäuser, Boston (1997)Google Scholar
  13. 13.
    Itoh, M., Satoh, H.: Information geometry of Poisson Kernels on Damek-Ricci spaces. Tokyo J. Math. 33, 129–144 (2010)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Itoh, M., Satoh, H.: Fisher information geometry, Poisson kernel and asymptotical harmonicity. Differ. Geom. Appl. 29, S107–S115 (2011)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Itoh, M., Satoh, H.: Fisher Information geometry of barycenter of probability measures, an oral presentaion at Geometric Sciences of Information, Paris (2013)Google Scholar
  16. 16.
    Itoh, M., Satoh, H.: Information geometry of Busemann-barycenter of probability measures, submittedGoogle Scholar
  17. 17.
    Itoh, M., Satoh, H., Suh, Y. J.: Horospheres and hyperbolicity of Hadamard manifold. Differ. Geom. Appl. 35, Supplement, 50–68 (2014)Google Scholar
  18. 18.
    Itoh, M., Shishido, Y.: Fisher information metric and Poisson kernels. Differ. Geom. Appl. 26, 347–356 (2008)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Ledrappier, F.: Harmonic measures and Bowen-Margulis measures. Israel J. Math. 71, 275–287 (1990)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. International Press, Boston (1994)MATHGoogle Scholar
  21. 21.
    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. AMS, Providence (2003)Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of TsukubaTsukubaJapan
  2. 2.Nippon Institute of TechnologySaitamaJapan

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