Information geometry of divergence functions

TLDR: This article studies the differential-geometrical structure of a manifold induced by a divergence function, which consists of a Riemannian metric, and a pair of dually coupled affine connections, which are studied in information geometry.

Abstract: 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 induced by a divergence function. It consists of a Riemannian metric, and a pair of dually coupled affine connections, which are studied in information geometry. The class of Bregman divergences are characterized by a dually flat structure, which is originated from the Legendre duality. A dually flat space admits a generalized Pythagorean theorem. The class of f -divergences, defined on a manifold of probability distributions, is characterized by information monotonicity, and the Kullback-Leibler divergence belongs to the intersection of both classes. The f -divergence always gives the α-geometry, which consists of the Fisher information metric and a dual pair of ±α-connections. The α-divergence is a special class of f -divergences. This is unique, sitting at the intersection of the f -divergence and Bregman divergence classes in a manifold of positive measures. The geometry derived from the Tsallis q-entropy and related divergences are also addressed.