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

Vladimir Jacimovic, Marijan Markovic

TL;DR

The paper addresses the need for group-invariant probabilistic models on hyperbolic spaces to support learning with non-Euclidean data. It introduces the conformally natural family F_{α,a} on the hyperbolic disc, with densities that are invariant under Möbius transformations and parameterized by α>1 and a in the unit disc, along with a practical random variate generation method and a tractable α=2 case. It then links these densities to mathematical physics as hyperbolic coherent states of SU(1,1), showing they are eigenfunctions of the hyperbolic Laplace-Beltrami operator and interpreting their mean under group action. The proposed framework offers a symmetry-preserving, scalable approach for probabilistic modeling in geometric deep learning and network science, with clear paths to higher-dimensional extensions and broader applications in ML and bioinformatics.

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.

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

TL;DR

The paper addresses the need for group-invariant probabilistic models on hyperbolic spaces to support learning with non-Euclidean data. It introduces the conformally natural family F_{α,a} on the hyperbolic disc, with densities that are invariant under Möbius transformations and parameterized by α>1 and a in the unit disc, along with a practical random variate generation method and a tractable α=2 case. It then links these densities to mathematical physics as hyperbolic coherent states of SU(1,1), showing they are eigenfunctions of the hyperbolic Laplace-Beltrami operator and interpreting their mean under group action. The proposed framework offers a symmetry-preserving, scalable approach for probabilistic modeling in geometric deep learning and network science, with clear paths to higher-dimensional extensions and broader applications in ML and bioinformatics.

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.
Paper Structure (8 sections, 6 theorems, 25 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The wrapped Cauchy family on the circle
  3. Conformally natural family of probability distributions over the hyperbolic disc
  4. Random variate generation in the hyperbolic disc
  5. The special case $\alpha = 2$
  6. Relations with mathematical physics
  7. Significance for ML and bioinformatics
  8. Conclusion

Key Result

Proposition 2.1

The family $wC(a)$ is invariant under the actions of the Möbius group ${\mathbb G}$. All wrapped Cauchy distributions are obtained as Möbius transformations of the uniform distribution on the circle.

Figures (2)

  • Figure 1: The plot of $100$ randomly generated points from $Z \sim {\mathcal{F}}_{a,\alpha}$ for $\alpha=2$ and $a=0$ (left), $a=1/2$ (right).
  • Figure 2: The plot of $100$ randomly generated points from $Z \sim {\mathcal{F}}_{a,\alpha}$ for $\alpha=10$ and $a=0$ (left), $a=1/2$ (right).

Theorems & Definitions (11)

  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • ...and 1 more