On the f-divergences between hyperboloid and Poincaré distributions

Author

Nielsen, Frank and Okamura, Kazuki

Abstract

Hyperbolic geometry has become popular in machine learning due to its capacity to embed discrete hierarchical graph structures with low distortions into continuous spaces for further downstream processing. It is thus becoming important to consider statistical models and inference methods for data sets grounded in hyperbolic spaces. In this work, we study the statistical f-divergences between two kinds of hyperbolic distributions: The Poincaré distributions and the related hyperboloid distributions. By exhibiting maximal invariants of group actions, we show how these f-divergences can be expressed as functions of canonical terms.