发表机构
Facultad de Ingeniería; Universidad de la República（工程学院; 共和国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对双曲空间嵌入在大半径下梯度学习失效的问题，提出用欧几里得预条件子重排布局、中间预条件子细化布局的两阶段优化方法，在真实树上损失降低46-74%。
AI 中文摘要
由于负曲率，层次图嵌入双曲空间比嵌入欧几里得空间具有更低的失真。然而，在大半径处，其基于梯度的学习会受到阻碍，此时庞加莱球模型和洛伦兹双曲面模型在数值上失效。极坐标避免了这一问题，但双曲度量将角步长按半径的双曲正弦进行缩放，从而冻结了角向运动。我们观察到，这一因子是一种选择，由现有实现悄然固定：例如，欧几里得切参数化使用半径本身。我们表明，其他选择不仅是可能的，而且更可取。它们是曲率从-1到0的单参数族优化预条件子的端点，而嵌入仍保持曲率-1。我们证明，由于欧几里得预条件子能重新排列布局但细化效果不佳，而中间预条件子在布局就位后细化效果好得多，因此将两者分两阶段结合，在真实世界树上比最佳单一曲率将损失降低46-74%。
英文摘要
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.
Comments7 pages, 2 figures
