We present the main features of the mathematical theory generated by the κ-deformed
exponential function exp κ (x)=(1+ κ 2 x 2+ κ x) 1/κ, with 0≤ κ< 1, developed in the last …

This is the first comprehensive book on information geometry, written by the founder of the
field. It begins with an elementary introduction to dualistic geometry and proceeds to a wide …

The axiomatic structure of the κ-statistcal theory is proven. In addition to the first three
standard Khinchin–Shannon axioms of continuity, maximality, and expansibility, two further …

An information-geometrical foundation is established for the deformed exponential families
of probability distributions. Two different types of geometrical structures, an invariant …

We illustrate some financial applications of the Tsallis and Kaniadakis deformed
exponential. The minimization of the corresponding deformed divergence is discussed as a …

The differential-geometric structure of the set of positive densities on a given measure space
has raised the interest of many mathematicians after the discovery by CR Rao of the …

We study the Voronoi diagrams of a finite set of Cauchy distributions and their dual
complexes from the viewpoint of information geometry by considering the Fisher-Rao …

The link with exponential families has allowed k-means clustering to be generalized to a
wide variety of data-generating distributions in exponential families and clustering …

We review a nonparametric version of Amari's information geometry in which the set of
positive probability densities on a given sample space is endowed with an atlas of charts to …

Nowozin\textit {et al} showed last year how to extend the GAN\textit {principle} to all $ f $-
divergences. The approach is elegant but falls short of a full description of the supervised …