arXiv：cs.LG· Xingrun Li, Sho Kuno, Yusuke Mukuta, Xin Yang, Tatsuya Harada·· 4 小时前AI 评分35
双曲表示学习对多分类数据的隐式偏好：Busemann 风险视角
The Implicit Bias of Hyperbolic Representation Learning for Multiclass Data: A Busemann Risk Perspective
AI 导读
研究双曲空间多分类中固定类原型的黎曼梯度流的隐式偏好，框架涵盖交叉熵等一般置换不变相对间隔（PERM）损失。在大半径下，到各原型的距离分解为径向项与由 Busemann 函数描述的方 向项，据此证明径向二分：漂移系数 μ 的符号决定半径趋向理想边界还是返回内部，正漂移持续时 r(t)=½log t+O(1)，负漂移则在有限时间内回到大半径阈值。
正文
Abstract: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 $\mu$ 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.
| Comments: | Accepted at NeurIPS 2026 as a Spotlight |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.07131 [cs.LG] |
| (or arXiv:2610.07131v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.07131 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Xingrun Li [view email]
[v1]
Mon, 5 Oct 2026 17:58:08 UTC (1,330 KB)
来源：arXiv：cs.LG · arxiv.org