# A group-theoretic framework for machine learning in hyperbolic spaces

arXiv: https://arxiv.org/abs/2501.06934 · HTML: https://mlsanity.com/paper/2501.06934/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 58 references that are in the map.
- First posted on arXiv 12 Jan 2025.

## Builds on (most cited first, 28 of 28)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Poincar'e GloVe: Hyperbolic Word Embeddings](https://mlsanity.com/paper/1810.06546/) (2018)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Hyperbolic Attention Networks](https://mlsanity.com/paper/1805.09786/) (2018)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Differentiating through the Fr'echet Mean](https://mlsanity.com/paper/2003.00335/) (2020)
- [HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections](https://mlsanity.com/paper/2106.03306/) (2021)
- [Learning the Irreducible Representations of Commutative Lie Groups](https://mlsanity.com/paper/1402.4437/) (2014)
- [Hyperbolic Busemann Learning with Ideal Prototypes](https://mlsanity.com/paper/2106.14472/) (2021)
- [Hyperbolic Deep Reinforcement Learning](https://mlsanity.com/paper/2210.01542/) (2022)
- [Machine learning meets network science: dimensionality reduction for fast and efficient embedding of networks in the hyperbolic space](https://mlsanity.com/paper/1602.06522/) (2016)
- [Skip-gram word embeddings in hyperbolic space](https://mlsanity.com/paper/1809.01498/) (2018)
- [Scalable Hyperbolic Recommender Systems](https://mlsanity.com/paper/1902.08648/) (2019)
- [Neural Distance Embeddings for Biological Sequences](https://mlsanity.com/paper/2109.09740/) (2021)
- [The Geometry of Continuous Latent Space Models for Network Data](https://mlsanity.com/paper/1712.08641/) (2017)
- [A Rotated Hyperbolic Wrapped Normal Distribution for Hierarchical Representation Learning](https://mlsanity.com/paper/2205.13371/) (2022)
- [Hyperbolic Neural Collaborative Recommender](https://mlsanity.com/paper/2104.07414/) (2021)
- [Conformally Natural Families of Probability Distributions on Hyperbolic Disc with a View on Geometric Deep Learning](https://mlsanity.com/paper/2407.16733/) (2024)
- [Navigable maps of structural brain networks across species](https://mlsanity.com/paper/1801.06079/) (2018)

## Built on by (most cited first, 3 of 3)

- [Cartan Networks: Group theoretical Hyperbolic Deep Learning](https://mlsanity.com/paper/2505.24353/) (2025)
- [Busemann energy-based attention for emotion analysis in Poincar'e discs](https://mlsanity.com/paper/2604.06752/) (2026)
- [Information-Geometric Optimization on Spheres](https://mlsanity.com/paper/2606.07588/) (2026)

## Abstract (arXiv)

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.