Abstract

Information geometry has emerged from a study on invariant structure of a family of probability distributions. The invariance gives a second-order symmetric tensor $g$ and a third order-symmetric tensor $T$ as unique invariant quantities. A pair $(g, T)$ defines a Riemannian metric and dual affine connections which together preserves the metric. Information geometry studies a Riemannian manifold having a pair of dual affine connections. Such a structure also arises from an asymmetric divergence function and affine differential geometry. In particular, a dually flat Riemannian manifold is important for applications, because a generalized Pythagorean theorem and projection theorem hold. Wasserstein distance gives another important geometry which is non-invariant, preserving the metric of a sample space. We try to construct information geometry of the entropy-regularized Wasserstein distance.