A group-theoretic framework for machine learning in hyperbolic spaces

The idea of learning representations in hyperbolic spaces has rapidly gained prominence in the last decade, attracting a lot of attention and motivating extensive investigations. This rise of interest was partly launched by statistical-physical studies [1] which have shown that distinctive properties of complex networks are naturally preserved in negatively curved continuous spaces. Since complex networks are ubiquitous in modern science and everyday life, this relation with hyperbolic geometry provided a valuable hint for low-dimensional representations of hierarchical data [2]. More generally, structural information of any hierarchical data set may be better represented in negatively curved manifolds rather than in flat ones. This further implies that hyperbolic geometry provides a suitable framework for simultaneous learning of hypernymies, similarities and analogies. This hypothesis triggered the interest of many data scientists and machine learning (ML) researchers in hyperbolic geometry. Nowadays, hyperbolic ML is a rapidly developing young subdiscipline within the broader field of geometric deep learning [3].