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. When a manifold is dually flat (it may be curved in the sense of the Levi-Civita connection), we have a canonical divergence and a pair of convex functions from which the original dual geometry is reconstructed. The generalized Pythagorean theorem and projection theorem hold in such a manifold. This structure has lots of applications in information sciences including statistics, machine learning, optimization, computer vision and Tsallis statistical mechanics. The present article reviews the structure of information geometry and its relation to the divergence function. We further consider the conformal structure given rise to by the generalized statistical model in relation to the power law.