A group-theoretic framework for machine learning in hyperbolic spaces
This work addresses the need for more principled approaches in hyperbolic ML, which is incremental as it builds upon existing methods to enhance foundations for designing algorithms like hyperbolic deep learning pipelines.
The paper tackles the problem of improving mathematical foundations for machine learning in hyperbolic spaces by introducing a group-theoretic framework, resulting in the development of a mean (barycenter) concept, novel probability distributions, and efficient optimization algorithms for computation and maximum likelihood estimation.
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.