Hyperbolic Graph Representation Learning: Embed in One Metric, Optimize with Another

2026-10-05

AI Summary

This study addresses the numerical instability and angle collapse encountered during gradient-based optimization of hyperbolic graph embeddings by proposing a family of preconditioners based on Euclidean tangent parameterization. Methodologically, the metric scaling factor is revealed as an optional design choice, enabling the construction of a preconditioner family whose curvature varies continuously from −1 to 0, thereby overcoming the limitations of a single fixed metric. Furthermore, a two-stage strategy combining distinct curvatures decouples low-distortion embedding from efficient optimization. Experimental results demonstrate that the proposed approach reduces the loss by 46% to 74% compared to the best single-curvature baseline on real-world tree-structured data, significantly improving representation quality.

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.

Problem

• Hyperbolic Graph Representation Learning
• Gradient-based Optimization
• Hierarchical Graphs
• Numerical Instability
• Optimization Preconditioner

Innovation

• Hyperbolic Graph Representation Learning
• Optimization Preconditioner
• Two-stage Optimization
• Hierarchical Graphs
• Curvature Family