Guest Speaker

Shun-ichi Amari,  RIKEN Brain Science Institute, Japan

Information Geometry and Its Applications:  Survey

Abstract

 

Information geometry emerged from the study of the geometrical structure of a manifold of probability distributions under the criterion of invariance.  It defines a Riemannian metric uniquely, which is the Fisher information metric.  Moreover, a family of dually coupled affine connections are introduced.  Mathematically, this is a study of a triple {M, g, T}, where M is a manifold, g is a Riemannian metric, and T is a third-order symmetric tensor.  Information geometry has been applied not only to statistical inferences but also to various fields of information sciences where probability plays an important role.

Many important families of probability distributions are dually flat Riemannian manifolds.  A dually flat manifold possesses a beautiful structure:  It has two mutually coupled flat affine connections and two convex functions connected by the Legendre transformation.  It has a canonical divergence, from which all the geometrical structure is derived.  The KL-divergence in probability distributions is automatically derived from the invariant flat nature.  Moreover, the generalized Pythagorean and geodesic projection theorems hold.

Conversely, we can define a dually flat Riemannian structure from a convex function.  This is derived through the Legendre transformation and Bregman divergence connected with a convex function.  Therefore, information geometry is applicable to convex analysis, even when it is not connected with probability distributions.  This widens the applicability of information geometry to convex analysis, machine learning, computer vision, Tsallis entropy, economics, and game theory.

The present talk summarizes theoretical constituents of information geometry and surveys a wide range of its applications.

References
  • S. Amari and H. Nagaoka, Methods of Information Geometry, American Mathematical Society and Oxford University Press, 2000

  •  

    S. Amari, Information geometry and its applications: Convex function and dually flat manifold.  Emerging Trends in Visual Computing, edited by F. Nielsen, Springer Lecture Notes in Computer Science, vol. 5416, pp 75-102, 2009

Biography

Shun-ichi Amari  received Dr. Eng. degree from the University of Tokyo in 1963.  He had worked as a professor at the University of Tokyo and is now Professor-Emeritus.  He served as Director of RIKEN Brain Science Institute for five years, and is now its senior advisor.   He is a foreign member of the Polish Academy of Science.  He has been engaged in research in wide areas of mathematical engineering, in particular, mathematical foundations of neural networks, including statistical neurodynamics, dynamical theory of neural fields, associative memory, self-organization, and general learning theory.  Another main subject of his research is information geometry initiated, which provides a new powerful method to information sciences.  Dr. Amari served as President of International Neural Networks Society and President of Institute of Electronics, Information and Communication Engineers, Japan.  He received Emanuel A. Piore Award and Neural Networks Pioneer Award from IEEE, the Japan Academy Award, Order of Cultural Merit of Japan, Gabor Award, Caianiello Award, Bosom Friend Award from Chinese Neural Networks Council, and C&C award, among many others.

http://www.brain.riken.jp/labs/mns/amari/home-E.html