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A group-theoretic framework for machine learning in hyperbolic spaces

Vladimir Jaćimović

TL;DR

The paper addresses principled learning in hyperbolic spaces by building a group-theoretic and conformal-geometric foundation for hyperbolic representations. It introduces conformal barycenters in the Poincaré disc and ball, holomorphic barycenters in Bergman balls, and Möbius as well as holomorphically natural probability models, all with conformal or holomorphic invariance. Efficient gradient-flow swarms are developed to compute barycenters and perform maximum-likelihood estimation in these spaces, together with explicit random-variate generation procedures. This framework enables principled, geometry-aware ML pipelines in hyperbolic latent spaces and provides a solid mathematical basis for extending probabilistic modeling and inference in hyperbolic domains.

Abstract

Embedding the data in hyperbolic spaces can preserve complex relationships in very few dimensions, thus enabling compact models and improving efficiency of machine learning (ML) algorithms. The underlying idea is that hyperbolic representations can prevent the loss of important structural information for certain ubiquitous types of data. However, further advances in hyperbolic ML require more principled mathematical approaches and adequate geometric methods. The present study aims at enhancing mathematical foundations of hyperbolic ML by combining group-theoretic and conformal-geometric arguments with optimization and statistical techniques. Precisely, we introduce the notion of the mean (barycenter) and the novel family of probability distributions on hyperbolic balls. We further propose efficient optimization algorithms for computation of the barycenter and for maximum likelihood estimation. One can build upon basic concepts presented here in order to design more demanding algorithms and implement hyperbolic deep learning pipelines.

A group-theoretic framework for machine learning in hyperbolic spaces

TL;DR

The paper addresses principled learning in hyperbolic spaces by building a group-theoretic and conformal-geometric foundation for hyperbolic representations. It introduces conformal barycenters in the Poincaré disc and ball, holomorphic barycenters in Bergman balls, and Möbius as well as holomorphically natural probability models, all with conformal or holomorphic invariance. Efficient gradient-flow swarms are developed to compute barycenters and perform maximum-likelihood estimation in these spaces, together with explicit random-variate generation procedures. This framework enables principled, geometry-aware ML pipelines in hyperbolic latent spaces and provides a solid mathematical basis for extending probabilistic modeling and inference in hyperbolic domains.

Abstract

Embedding the data in hyperbolic spaces can preserve complex relationships in very few dimensions, thus enabling compact models and improving efficiency of machine learning (ML) algorithms. The underlying idea is that hyperbolic representations can prevent the loss of important structural information for certain ubiquitous types of data. However, further advances in hyperbolic ML require more principled mathematical approaches and adequate geometric methods. The present study aims at enhancing mathematical foundations of hyperbolic ML by combining group-theoretic and conformal-geometric arguments with optimization and statistical techniques. Precisely, we introduce the notion of the mean (barycenter) and the novel family of probability distributions on hyperbolic balls. We further propose efficient optimization algorithms for computation of the barycenter and for maximum likelihood estimation. One can build upon basic concepts presented here in order to design more demanding algorithms and implement hyperbolic deep learning pipelines.
Paper Structure (19 sections, 16 theorems, 97 equations, 4 figures)

This paper contains 19 sections, 16 theorems, 97 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on hyperbolic balls
  3. Poincaré disc
  4. Poincaré ball
  5. Bergman ball
  6. Conformal barycenter in the Poincaré disc
  7. Definition and properties
  8. Computation of the conformal barycenter
  9. Statistical modeling in the Poincaré disc
  10. The Möbius family of probability distributions in the Poincaré disc
  11. Generation of random points in the Poincaré disc
  12. Maximum likelihood for the Möbius family
  13. Barycenters in hyperbolic balls
  14. Conformal barycenter in the Poincaré ball
  15. Holomorphic barycenter in the Bergman ball
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\{z_1,\dots,z_N\}$ be a configuration of points in $\mathbb{B}^2$. Then, there exists a unique (up to a rotation) Möbius transformation $g_a \in \mathbb{G}_2$, such that the configuration $\{g_a(z_1),\dots,g_a(z_N)\}$ is balanced.

Figures (4)

  • Figure 1: This Figure illustrates balanced configurations corresponding to configurations consisting of (a) three points; (b) four points and (c) five points.
  • Figure 2: Barycenters in the Poincaré disc.
  • Figure 3: Random samples in the Poincaré disc with parameters: a) $a=0$, $s=2$; b) $a=0$, $s=4$; c) $a=0$, $s=8$; d) $a=0.9e^{i\frac{3\pi}{4}}$, $s=2$; e) $a=0.9e^{i\frac{3\pi}{4}}$, $s=4$; f) $a=0.9e^{i\frac{3\pi}{4}}$, $s=8$.
  • Figure 4: Random samples from the Poincare ball $\mathbb{B}^3$ for parameter values: a) $a=(0,0,0)$, $s=3$; b) $a=(0,0,0)$, $s=5$; c) $a=(0.9,0,0)$, $s=3$; d) $a=(0.9,0,0)$, $s=5$.

Theorems & Definitions (30)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Definition 4
  • Proposition 2
  • Proposition 3
  • Theorem 4
  • Theorem 5
  • ...and 20 more