Geometry from divergence functions and complex structures

Differential Geometry Derived from Divergence Functions: Information Geometry Approach

We study differential-geometrical structure of an information manifold equipped with a divergence function. A divergence function generates a Riemannian metric and furthermore it provides a symmetric third-order tensor, when the divergence is asymmetric. This induces a pair of affine connections dually coupled to each other with respect to the Riemannian metric. This is the arising emerged from...

Information geometry of divergence functions

Measures of divergence between two points play a key role in many engineering problems. One such measure is a distance function, but there are many important measures which do not satisfy the properties of the distance. The Bregman divergence, KullbackLeibler divergence and f -divergence are such measures. In the present article, we study the differential-geometrical structure of a manifold ind...

Divergence Functions and Geometric Structures They Induce on a Manifold

Divergence functions play a central role in information geometry. Given a manifold M, a divergence function D is a smooth, nonnegative function on the product manifold M ×M that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold ∆M ⊂ M ×M. In this Chapter, we review how such divergence functions induce i) a statistical st...

Symplectic and Kähler Structures on Statistical Manifolds Induced from Divergence Functions

Divergence functions play a central role in information geometry. Given a manifold M, a divergence function D is a smooth, nonnegative function on the product manifold M×M that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold Mx. It is well-known (Eguchi, 1982) that the statistical structure on M (a Riemmanian metric wi...

GEOMETRY OF f-DIVERGENCE*