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.
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Notes
- 1.
Hyperbolic geometry has constant negative curvature and the volume of hyperbolic balls increases exponentially with respect to their radii rather than polynomially as in Euclidean spaces.
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Nielsen, F., Okamura, K. (2023). On the f-Divergences Between Hyperboloid and Poincaré Distributions. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_18
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