Conformally Natural Families of Probability Distributions on Hyperbolic Disc with a View on Geometric Deep Learning

TL;DR

This paper introduces a new family of probability distributions on hyperbolic discs that are invariant under conformal mappings, facilitating uncertainty modeling in hyperbolic data for applications like Geometric Deep Learning and bioinformatics.

Contribution

The paper presents a novel, conformally natural family of distributions on hyperbolic discs, enhancing modeling capabilities for hyperbolic data with invariance properties.

Findings

Distribution family is invariant under conformal mappings.

Potential applications in Geometric Deep Learning and bioinformatics.

Analogies with hyperbolic coherent states in quantum physics.

Abstract

We introduce the novel family of probability distributions on hyperbolic disc. The distinctive property of the proposed family is invariance under the actions of the group of disc-preserving conformal mappings. The group-invariance property renders it a convenient and tractable model for encoding uncertainties in hyperbolic data. Potential applications in Geometric Deep Learning and bioinformatics are numerous, some of them are briefly discussed. We also emphasize analogies with hyperbolic coherent states in quantum physics.