Semilocal properties of canonical divergences in dually flat spaces

  • Atsuya KumagaiEmail author
Original Paper Area 1


The properties of divergences in dually flat spaces are investigated beyond the local perspective. First, an approximative representation of divergence is shown on the basis of the expansion with respect to relative coordinates. A consideration concerning triangular relation is done from similar perspective and an equation which extends classical multidimensional scaling to incorporate Riemannian metric and affine connection coefficients is obtained. Furthermore, a consideration from a viewpoint of probability distribution is done. The representation of divergence can be regarded as Kullback–Leibler divergence between normal distributions with a common covariance matrix. Especially, it is shown that the asymmetry of divergence arises from the spatial change in differential entropy in those distributions.


Information geometry Divergence Dually flat space 

Mathematics Subject Classification

51K05 91C15 94A17 


  1. 1.
    Ackermann, M., Blömer, J., Scholz, C.: Hardness and non-approximability of Bregman clustering problems. Technical report, Electronic colloquium on computational complexity (2011).
  2. 2.
    Amari, S., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  3. 3.
    Banerjee, A., Merugu, S., Dhillon, I.S., Ghosh, J.: Clustering with Bregman divergences. J. Mach. Learn. Res. 6, 1705–1749 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bishop, C.: Pattern Recognition and Machine Learning. Springer, New York (2006)zbMATHGoogle Scholar
  5. 5.
    Borg, I., Groenen, P.: Modern Multidimensional Scaling, 2nd edn. Springer, New York (2005)zbMATHGoogle Scholar
  6. 6.
    Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brun, A., Knutsson, H.: Tensor Glyph Warping: Visualizing metric tensor fields using Riemannian exponential maps, In: Laidlaw, D. H., Weickert, J. (eds.) Visualization and Processing of Tensor Fields, pp. 139–160. Springer, Berlin Heidelberg (2009)Google Scholar
  8. 8.
    Chino, N., Shiraiwa, K.: Geometrical structures of some non-distance models for asymmetric MDS. Behaviormetrika 20, 35–47 (1993)CrossRefGoogle Scholar
  9. 9.
    Chino, N.: A brief survey of asymmetric MDS and some open problems. Behaviormetrika 39, 127–165 (2012)CrossRefGoogle Scholar
  10. 10.
    Cox, T.F., Cox, M.A.A.: Multidimensional Scaling, 2nd edn. Chapman and Hall/CRC, Boca Raton (2000)zbMATHGoogle Scholar
  11. 11.
    Escoufier, Y., Grorud, A.: Analyse factorielle des matrices carées non-symétriques. In: Diday, E. (ed.) Data Analysis and Informatics, pp. 263–276. North-Holland, Amsterdam (1980)Google Scholar
  12. 12.
    Krumhansl, C.L.: Concerning the applicability of geometric models to similarity data: the interrelationship between similarity and spatial density. Psychol. Rev. 85, 445–463 (1987)CrossRefGoogle Scholar
  13. 13.
    Kumagai, A.: Extension of classical MDS to treat dissimilarities not satisfying axioms of distance. Jpn. J. Ind. Appl. Math. 31, 111–124 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kumagai, A.: Multidimensional scaling in dually flat spaces. Jpn. J. Indust. Appl. Math. 32, 51–63 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    MacAdam, D.L.: Visual sensitivities to color differences in daylight. J. Opt. Soc. Am. 32, 247–273 (1942)CrossRefGoogle Scholar
  16. 16.
    Nielsen, F., Nock, R.: Clustering multivariate normal distributions. Lect. Notes Comput. Sci. 5416, 164–174 (2009)CrossRefGoogle Scholar
  17. 17.
    Okada, A., Imaizumi, T.: Nonmetric multidimensional scaling of asymmetric proximities. Behaviormetrika 21, 81–96 (1987)CrossRefGoogle Scholar
  18. 18.
    Saito, T., Yadohisa, H.: Data Analysis of Asymmetric Structures. Marcel Dekker, New York (2005)zbMATHGoogle Scholar
  19. 19.
    Torgerson, W.S.: Theory and Methods of Scaling. Wiley, New York (1958)Google Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2016

Authors and Affiliations

  1. 1.College of CommerceNihon UniversityTokyoJapan

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