双曲表示学习对多类数据的隐式偏差：Busemann风险视角
The Implicit Bias of Hyperbolic Representation Learning for Multiclass Data: A Busemann Risk Perspective
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中文总结 AI 辅助
本研究从Busemann风险视角分析了双曲多类分类中黎曼梯度流的隐式偏差，揭示了径向二分法和边界方向收敛性，解释了边界饱和与近边界聚类现象。
中文摘要 AI 辅助
我们研究了在双曲空间$\mathbb{H}^n$中，具有固定类原型的双曲多类分类的黎曼梯度流的隐式偏差。我们的框架适用于一般的置换不变相对边际（PERM）损失，该类损失包括交叉熵和其他标准多类损失。我们的分析基于一个分解：在大半径处，到每个原型的距离分解为一个径向项和一个由Busemann函数描述的依赖于方向项。这产生了两个主要结果。首先，我们证明了径向二分法：漂移系数$\mu$的符号决定了半径是被推向理想边界还是被推回内部；如果正漂移持续存在，则$r(t)=\frac{1}{2}\log t+O(1)$，而持续负漂移则在有限时间内将轨迹返回到大半径阈值。其次，我们证明了边界方向收敛到$\partial\mathbb{H}^n$上Busemann风险的临界点。这些结果为双曲表示学习中的两种现象提供了严格的渐近视角，我们称之为边界饱和和近边界聚类。
英文摘要
We study the implicit bias of Riemannian gradient flow for hyperbolic multiclass classification with fixed class prototypes in hyperbolic space $\mathbb{H}^n$. Our framework accommodates general permutation invariant relative margin (PERM) losses, a class that includes cross entropy and other standard multiclass losses. Our analysis is based on a decomposition: at large radius, the distance to each prototype splits into a radial term and a direction-dependent term described by the Busemann function. This yields two main results. First, we prove a radial dichotomy: the sign of a drift coefficient $μ$ determines whether the radius is pushed toward the ideal boundary or back toward the interior; if the positive drift persists, then $r(t)=\frac{1}{2}\log t+O(1)$, while persistent negative drift returns the trajectory to the large-radius threshold in finite time. Second, we show that the boundary direction converges to a critical point of the Busemann risk on $\partial\mathbb{H}^n$. These results provide a rigorous asymptotic perspective on two phenomena we refer to as boundary saturation and near-boundary clustering in hyperbolic representation learning.
发表机构
- The University of Tokyo（东京大学）
- RIKEN（理化学研究所）
机构由 AI 辅助整理，请以论文原文为准。
