Hyperbolic Graph Representation Learning: Embed in One Metric, Optimize with Another
Abstract
Hierarchical graphs embed in hyperbolic space with lower distortion than in Euclidean space owing to its negative curvature. However, their gradient-based learning is hampered at large radii, where the Poincaré ball and the Lorentz hyperboloid models fail numerically. Polar coordinates avoid this problem, but the hyperbolic metric scales the angular step by the hyperbolic sine of the radius, freezing angular motion. We observe that this factor is a choice, silently fixed by existing implementations: the Euclidean tangent parametrization, for instance, uses the radius itself. We show that other choices are not only possible but preferable. They are endpoints of a one-parameter family of optimization preconditioners with curvatures from −1 to $0$, while the embedding remains at curvature −1. We show that since the Euclidean preconditioner rearranges a layout but refines it poorly, while an intermediate one refines far better once a layout is in place, combining them in two stages reduces the loss on real-world trees by 46-74% over the best single curvature.
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BibTeX
@misc{larroca2026hyperbolic,
title = {{Hyperbolic Graph Representation Learning: Embed in One Metric, Optimize with Another}},
author = {Federico Larroca and Paola Bermolen and Marcelo Fiori and Bernardo Marenco},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06745},
eprint = {2610.06745},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06745},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Hyperbolic Graph Representation Learning: Embed in One Metric, Optimize with Another
AU - Federico Larroca
AU - Paola Bermolen
AU - Marcelo Fiori
AU - Bernardo Marenco
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06745
UR - https://arxiv.org/abs/2610.06745
AB - Hierarchical graphs embed in hyperbolic space with lower distortion than in Euclidean space owing to its negative curvature. However, their gradient-based learning is hampered at large radii, where the Poincaré ball and the Lorentz hyperboloid models fail numerically. Polar coordinates avoid this problem, but the hyperbolic metric scales the angular step by the hyperbolic sine of the radius, freezing angular motion. We observe that this factor is a choice, silently fixed by existing implementations: the Euclidean tangent parametrization, for instance, uses the radius itself. We show that other choices are not only possible but preferable. They are endpoints of a one-parameter family of optimization preconditioners with curvatures from −1 to $0$, while the embedding remains at curvature −1. We show that since the Euclidean preconditioner rearranges a layout but refines it poorly, while an intermediate one refines far better once a layout is in place, combining them in two stages reduces the loss on real-world trees by 46-74% over the best single curvature.
ER -