We explore the information geometric structure of the statistical manifold generated by the κ-deformed exponential family. The dually-flat manifold is obtained as a dualistic Hessian structure by introducing suitable generalization of the Fisher metric and affine connections. As a byproduct, we obtain the fluctuation-response relations in the κ-formalism based on the κ-generalized exponential f...

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...

Divergences, also known as contrast functions, are distance-like quantities defined on manifolds of non-negative or probability measures and they arise in various theoretical and applied problems. Using ideas in optimal transport, we introduce and study a parameterized family of $L^{(\pm \alpha)}$-divergences which includes the Bregman divergence corresponding to the Euclidean quadratic cost, a...

We investigate questions in quantum information geometry which concern the existence and non-existence of dual and dually flat structures on stratified sets of density operators on finite-dimensional Hilbert spaces. We show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal. We prove moreover ...

In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalr flat metric having constant mean curvature. This result can be seen as a generalization to higher dimensions of the well known Riemann mapping Theorem in the plane.

Let M be a complete locally compact CAT(0)-space, and X an ultralimit of M . For γ ⊂M a k-dimensional flat, let γω be the k-dimensional flat in X obtained as an ultralimit of γ. In this paper, we identify various conditions on γω that are sufficient to ensure that γ bounds a (k + 1)-dimensional half-flat. As applications we obtain (1) constraints on the behavior of quasi-isometries between loca...