Information Geometry and Its Applications:

Survey

Shun-ichi Amari

RIKEN Brain Science Institute, Japan

Abstract. Information geometry emerged from the study of the geo-metrical structure of a manifold of probability distributions under thecriterion of invariance. It defines a Riemannian metric uniquely, which isthe Fisher information metric. Moreover, a family of dually coupled affineconnections 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 isa third-order symmetric tensor. Information geometry has been appliednot only to statistical inferences but also to various fields of informationsciences where probability plays an important role. Many important fam-ilies of probability distributions are dually flat Riemannian manifolds. Adually flat manifold possesses a beautiful structure: It has two mutuallycoupled flat affine connections and two convex functions connected bythe Legendre transformation. It has a canonical divergence, from whichall the geometrical structure is derived. The KL-divergence in probabil-ity distributions is automatically derived from the invariant flat nature.Moreover, the generalized Pythagorean and geodesic projection theoremshold. Conversely, we can define a dually flat Riemannian structure froma convex function. This is derived through the Legendre transformationand Bregman divergence connected with a convex function. Therefore,information geometry is applicable to convex analysis, even when it is notconnected with probability distributions. This widens the applicabilityof information geometry to convex analysis, machine learning, computervision, Tsallis entropy, economics, and game theory. The present talksummarizes theoretical constituents of information geometry and sur-veys a wide range of its applications.

References

1. Amari, Nagaoka, H.: Methods of Information Geometry. American MathematicalSociety, Oxford University Press (2000)

2. Amari, S.: Information geometry and its applications: Convex function and du-ally flat manifold. In: Nielsen, F. (ed.) ETVC 2008. LNCS, vol. 5416, pp. 75–102.Springer, Heidelberg (2009)

F. Nielsen and F. Barbaresco (Eds.): GSI 2013, LNCS 8085, p. 3, 2013.c© Springer-Verlag Berlin Heidelberg 2013