Michel Deza , Michel Petitjean , Krassimir Markov

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 withrespect to the Riemannian metric. This is the arising emerged from information geometry. When a manifold is duallyflat (it may be curved in the sense of the Levi-Civita connection), we have a canonical divergence and a pair ofconvex functions from which the original dual geometry is reconstructed. The generalized Pythagorean theoremand projection theorem hold in such a manifold. This structure has lots of applications in information sciencesincluding statistics, machine learning, optimization, computer vision and Tsallis statistical mechanics. The presentarticle reviews the structure of information geometry and its relation to the divergence function. We further considerthe conformal structure given rise to by the generalized statistical model in relation to the power law.