Advertisement

A perturbative picture of cubic tensors in dually flat spaces

  • Atsuya KumagaiEmail author
Original Paper Area 1
  • 54 Downloads

Abstract

Some effects of cubic tensors in divergences, which characterize dually flat spaces, are investigated. Focusing on the small area around a specific point, how tangent space approximation is corrected by the cubic tensor at the point, is described. The cubic tensor in divergence is treated as a perturbation, supposing that the coordinates in tangent space are unperturbed solutions. A first order perturbation solution is presented as a main result. On the other hand, the unperturbed solution is yet another coordinate different from the usual affine coordinate. The divergences are shown to be written in a simple form if it is written in terms of the unperturbed solution. Furthermore, a relation which transforms similarities to the divergences is shown. Consequently, the asymmetry in the divergences between the tangent point and the other point is shown to be attributed to the difference between the squared norms of the two kinds of coordinates.

Keywords

Information geometry Divergence Dually flat space Multidimensional scaling 

Mathematics Subject Classification

53A15 91C15 94A17 

Notes

Acknowledgements

The author greatly appreciates the valuable comments of the reviewers.

References

  1. 1.
    Amari, S., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  2. 2.
    Banerjee, A., Merugu, S., Dhillon, I.S., Ghosh, J.: Clustering with Bregman divergences. J. Mach. Learn. Res. 6, 1705–1749 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Borg, I., Groenen, P.: Modern Multidimensional Scaling, 2nd edn. Springer, New York (2005)zbMATHGoogle Scholar
  4. 4.
    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
  5. 5.
    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 (2009)CrossRefGoogle Scholar
  6. 6.
    Kumagai, A.: Multidimensional scaling in dually flat spaces. Jpn. J. Ind. Appl. Math. 32, 51–63 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kumagai, A.: Semilocal properties of canonical divergences in dually flat spaces. Jpn. J. Ind. Appl. Math. 33, 417–426 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lee, J.A., Verleysen, M.: Nonlinear Dimensionality Reduction. Springer, New York (2010)zbMATHGoogle Scholar
  9. 9.
    MacAdam, D.L.: Visual sensitivities to color differences in daylight. J. Opt. Soc. Am. 32, 247–273 (1942)CrossRefGoogle Scholar
  10. 10.
    Pekalska, E., Paclík, P., Duin, R.P.W.: A generalized kernel approach to dissimilarity-based classification. J. Mach. Learn. Res. 2, 175–211 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Schölkopf, B.: The kernel trick for distances. In: Neural Information Processing Systems (2000)Google Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK 2017

Authors and Affiliations

  1. 1.College of CommerceNihon UniversityTokyoJapan

Personalised recommendations