🤖 AI Summary
This work investigates the construction of a divergence measure between Gaussian distributions in hyperbolic space that remains invariant under Möbius transformations. Building upon the Poincaré disk model, the study establishes a geometric duality between an L² embedding and the spherical squared Hellinger distance, thereby uncovering an intrinsic connection between this divergence and hyperbolic isometric invariants. Leveraging this insight, the authors propose a novel Gaussian divergence that is rigorously invariant under the action of the Möbius group. This contribution not only ensures robustness under hyperbolic isometries but also offers a fresh perspective and new geometric tools for information-theoretic divergence analysis by integrating principles from hyperbolic geometry.
📝 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.