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This research addresses numerical instability in hyperbolic graph embeddings by introducing a tunable family of optimization preconditioners that decouple the embedding curvature from the metric scaling factor. By combining Euclidean rearrangement with intermediate refinement stages, the proposed method significantly reduces training loss on hierarchical datasets compared to standard single-curvature approaches. The work advances the practical viability of hyperbolic geometry for representing large-scale tree-like structures in machine learning models.
Hyperbolic Graph Representation Learning: Embed in One Metric, Optimize with Another
arXiv:2610.06745v1 Announce Type: new 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 Poincar\'e 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.
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.
Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2610.06745 [cs.LG]
(or arXiv:2610.06745v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.06745 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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