On the geometry of generalized Gaussian distributions

TL;DR

This paper explores the geometric structure of the space of distributions maximizing q-Rényi entropy, unifying various metrics, and analyzing their curvature and geodesics to understand statistical distinguishability.

Contribution

It introduces a unified Riemannian metric covering multiple classical and quantum information metrics, and computes their geometric properties and implications.

Findings

Derived the geodesic equations and solutions for the unified metric.

Calculated the Riemann, Ricci, and scalar curvatures of the metric.

Showed how the parameter q influences the distinguishability of distributions.

Abstract

In this paper we consider the space of those probability distributions which maximize the $q$-R\'enyi entropy. These distributions have the same parameter space for every $q$, and in the $q=1$ case these are the normal distributions. Some methods to endow this parameter space with Riemannian metric is presented: the second derivative of the $q$-R\'enyi entropy, Tsallis-entropy and the relative entropy give rise to a Riemannian metric, the Fisher-information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovri\'c, Min-Oo and Ruh. These metrics are different therefore our differential geometrical calculations based on a unified metric, which covers all the above mentioned metrics among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors and scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter $q$ modulates the statistical distinguishability of close points. We show that some frequently used metric in quantum information geometry can be easily recovered from classical metrics.