A group-theoretic framework for machine learning in hyperbolic spaces

TL;DR

This paper develops a rigorous mathematical framework for hyperbolic space machine learning by introducing group-theoretic and geometric methods, including new probability distributions and optimization algorithms, to improve model efficiency and structural preservation.

Contribution

It introduces a formal group-theoretic and conformal geometric foundation for hyperbolic ML, including new probability distributions and optimization techniques.

Findings

Defined the hyperbolic barycenter and probability distributions

Developed efficient algorithms for barycenter computation

Enhanced mathematical tools for hyperbolic deep learning

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.