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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
Machine learning needs probability models for uncertainty in curved data, including hyperbolic spaces. This paper introduces a Möbius-invariant family on the hyperbolic disc, with invariance supporting implementation and transparency in stochastic policies.
Problem
Geometry-informed machine-learning models are needed to address data symmetries and inherent curvature, including uncertainty in hyperbolic spaces.
Source: §1 · p. 1, §6 · p. 10Method
The paper introduces a probability family on the hyperbolic disc invariant under the actions of its isometry group.
Source: §1 · p. 2, §6 · p. 11Results
The proposed family is invariant under the Möbius group, which acts on fixed-α subfamilies rather than transitively on the whole class.
Source: §3 · p. 5, §3.2 · p. 8Takeaways & Limitations
Learning over group-invariant families reduces to learning group orbits and infinitesimal generators, facilitating stochastic-policy implementation and algorithm transparency.
Source: §6 · p. 11The proposed model is for hyperbolic discs; extensions to higher-dimensional hyperbolic balls or multidiscs could be significant for Geometric DL.
Source: §6 · p. 11
Abstract
from arXiv · showhide
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.
1. Introduction
Gaussian models’ invariance makes them useful across continuous-space machine learning, but their flat-space setting does not cover data with nonzero curvature. The paper introduces a group-invariant probability family on the hyperbolic disc for probabilistic machine learning.
1. Introduction
- Gaussian models’ strong role in continuous-space learning is tied to invariance under linear-affine transformations, but their representative power is modest.Mixtures can address setups where that limitation matters.Source: §1 · p. 1
- Many scientific, engineering, and everyday datasets have nonzero curvature, including spherical orientation data and hyperbolic hierarchical data.Examples of hyperbolic data include complex networks and semantic hierarchies [3] [4].Source: §1 · pp. 1–2
- Uncertainty in curved spaces requires probability models over those spaces, while negative-curvature statistical models remain poorly investigated and outside directional statistics’ scope.The paper links this need to recent advances in machine learning over hyperbolic geometries.Source: §1 · p. 2
- The paper introduces a probability family on the hyperbolic disc whose group invariance is intended to support probabilistic machine learning.The authors state that this property facilitates algorithm implementation and improves transparency.Source: §1 · p. 2
2. The wrapped Cauchy family on the circle
The wrapped Cauchy family is obtained by wrapping real-line Cauchy distributions onto the circle and has a hyperbolic-disc statistical geometry. Its Möbius invariance coexists with limited representational power, while related circular models offer different trade-offs.
2. The wrapped Cauchy family on the circle
- Wrapped Cauchy distributions result from Cayley-transform wrapping of real-line Cauchy distributions; the family includes the uniform density and approaches circle Dirac deltas as r tends to 1.The uniform case occurs at r = 0.Source: §2 · pp. 2–3
- The Fisher information metric on wC(a) coincides with the hyperbolic metric of the unit disc, making its statistical manifold isomorphic to that disc.The parameter a lies in the unit disc, and the density’s mean is a.Source: §2 · pp. 2–3
- All wrapped Cauchy distributions arise as Möbius transforms of the uniform circle distribution, and the family is invariant under the disc-preserving Möbius group.The group acts transitively on the family.Source: §2 · p. 3
- The wrapped Cauchy family has limited representative power because its densities are unimodal and symmetric, though mixtures can improve that power.Source: §2 · p. 3
- Unlike wrapped Cauchy, the von Mises family is invariant only under planar rotations; Kato-Jones has higher representative power and includes asymmetric and bimodal densities.Wrapped Cauchy is a special case of Kato-Jones when the initial von Mises distribution is uniform.Source: §2 · pp. 3–4
3. Conformally natural family of probability distributions over the hyperbolic disc
The paper defines a probability family on the hyperbolic disc that is invariant under Möbius transformations, with transitive fixed-α subfamilies. It also gives results on the family’s center of mass, sampling, and the especially tractable α = 2 case.
3. Conformally natural family of probability distributions over the hyperbolic disc
- The family Fα,a is invariant under the Möbius group, and for fixed α the group acts transitively on its two-dimensional subfamily.The full family is three-dimensional, parametrized by α > 1 and a in the unit disc.Source: §3 · p. 5
- The general formula for disk probabilities involves an integral over a Gauss hypergeometric series, while simpler expressions are available in special cases, notably α = 2.Source: §3 · pp. 5–6
3.1. Random variate generation in the hyperbolic disc.
- The distributions become more concentrated as α increases, with Dirac delta distributions arising in the limit α → ∞.The figures compare samples for α = 2 and α = 10 at different values of a.Source: §3 · p. 6, Figure 1 · p. 7, Figure 2 · p. 7
3. Conformally natural family of probability distributions over the hyperbolic disc
- The Riemannian center of mass of each density in (3.2) is its parameter a.Source: §3 · p. 7
3.1. Random variate generation in the hyperbolic disc.
- For fixed α, random samples can be generated from the a = 0 distribution and mapped by a Möbius transformation to obtain samples for parameter a.At a = 0, angles are uniform and radii are sampled by inverting the radial distribution function.Source: §3.1 · p. 8
3.2. The special case α = 2.
- The F2,a subfamily is a two-dimensional model with simple, tractable formulas for sampling and parameter estimation.It is presented for inference, black-box optimization, and probabilistic modeling on the hyperbolic disc.Source: §3.2 · pp. 8–9
- F2,a has limited representative power and cannot approach interior Dirac deltas; policies needing such convergence should increase α.Mixtures are suggested to address limited representative power.Source: §3.2 · pp. 8–9
4. Relations with mathematical physics
The paper connects the proposed densities to hyperbolic harmonic analysis and SU(1, 1) coherent states. This correspondence also suggests an analogy between learning over the family and quantum evolution controlled by a Hamiltonian.
4. Relations with mathematical physics
- Wrapped-Cauchy densities also appear as Poisson kernels, which are harmonic for the hyperbolic Laplace–Beltrami operator and central to complex analysis and potential theory.Source: §4 · p. 9
- Densities (3.2) are eigenfunctions of the hyperbolic Laplace–Beltrami operator with eigenvalues α(α −1), a property linked to their relevance in quantum optics.Source: §4 · p. 9
- The densities are SU(1, 1) coherent states in analytic representations, also called hyperbolic coherent states.Source: §4 · p. 9
- The density with a = 0 is the ground state; SU(1, 1) actions generate the family, parametrized by points in the hyperbolic disc.At a = 0 the densities are rotationally symmetric under SO(2), and SU(1, 1)/SO(2) = B2.Source: §4 · p. 9
- Learning over the family is analogous to quantum evolution on an invariant manifold of coherent states, and can be implemented by controlling a system Hamiltonian.The Hamiltonian is a linear combination of SU(1, 1) generators, so control updates its coefficients.Source: §4 · p. 9
5. Significance for ML and bioinformatics
The paper presents the family as a tractable framework for learning with uncertainty in hyperbolic data, including network and hierarchical-data applications. These uses are prospective: experiments in the cited learning settings remain to be conducted.
5. Significance for ML and bioinformatics
- The family is proposed as a framework for learning algorithms that handle uncertainty in hyperbolic data, including mixtures or generalizations.Potential settings include stochastic policies in reinforcement learning and evolutionary optimization over hyperbolic (multi)discs.Source: §5 · p. 10
- Hyperbolic network embeddings motivate potential applications in statistical network science and machine learning over hierarchical data.The paper notes that large real-world networks often obey a power law, which it connects to hyperbolic geometry.Source: §5 · p. 10
- Because Möbius-related point sets represent the same network, a fixed-α subfamily can represent that network; mixtures can model networks and support comparison, distance estimation, and subnetwork matching.Source: §5 · p. 10
- The paper reports one prior ML study using wrapped normal distributions on the hyperbolic disc, but describes that model as inconvenient and difficult to learn over.The cited study applied it to reinforcement learning, hyperbolic variational autoencoders, and probabilistic word embeddings.Source: §5 · p. 10
- Experiments with the proposed densities in supervised and unsupervised learning, and in reinforcement learning with hyperbolic state spaces, remain to be conducted.Source: §5 · p. 10
6. Conclusion
The study frames geometry-informed machine learning as needing statistical models on Riemannian manifolds, especially for negative-curvature data. It proposes a group-invariant probability family on hyperbolic discs and identifies higher-dimensional extensions as future work.
6. Conclusion
- Statistical models on Riemannian manifolds are needed to encode uncertainty in curved data, while directional statistics provides models for positive-curvature data.The paper characterizes directional statistics as fairly well understood, though not properly recognized and exploited by the ML community.Source: §6 · p. 11
- The proposed hyperbolic-disc probability family is invariant under hyperbolic isometries, and learning over group-invariant families reduces to learning orbits and infinitesimal generators.The authors say this facilitates stochastic-policy implementation and improves algorithm transparency.Source: §6 · p. 11
- The study targets negative-curvature data, which it describes as especially ubiquitous across science, engineering, and everyday life.Source: §6 · p. 11
- Higher-dimensional extensions may matter because large hierarchical datasets are embedded in hyperbolic balls or multidiscs.Source: §6 · p. 11
- Even-dimensional hyperbolic spaces above dimension two can have two hyperbolic metrics and corresponding isometry groups, yielding non-equivalent statistical models.Source: §6 · p. 11
- The authors identify group-invariant families on spheres and hyperbolic balls as directions for continued study.The cited discussion notes that wrapped Cauchy has already been extended to higher-dimensional spheres.Source: §6 · p. 11