Isometric Invariant Quantification of Gaussian Divergence over Poincare Disc

TL;DR

This paper introduces a hyperbolic isometric invariant based on the Poincare disc geometry to quantify divergence between Gaussian measures, leveraging a duality with spherical distances.

Contribution

It proposes a novel divergence measure for Gaussian distributions using hyperbolic geometry, expanding tools in information theory.

Findings

Establishes a geometric duality between spherical and hyperbolic distances.

Introduces an L2-embedded hyperbolic invariant for divergence measurement.

Provides a new geometric perspective on Gaussian divergence quantification.

Abstract

The paper presents a geometric duality between the spherical squared-Hellinger distance and a hyperbolic isometric invariant of the Poincare disc under the action of the general Mobius group. Motivated by the geometric connection, we propose the usage of the L2-embedded hyperbolic isometric invariant as an alternative way to quantify divergence between Gaussian measures as a contribution to information theory.