The Implicit Bias of Hyperbolic Representation Learning for Multiclass Data: A Busemann Risk Perspective

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 $μ$ determines whether the radius is pushed toward the ideal boundary or back toward the interior; if the positive drift persists, then $r(t)=\tfrac12\log t+O(1)$, while persistent negative drift returns the trajectory to the large radius threshold in finite time. Second, we show that large-radius directions can approach critical points of the Busemann risk on $\partial\mathbb H^n$, even during radial return. These results provide a rigorous large-radius perspective on two phenomena we refer to as boundary saturation and near-boundary clustering in hyperbolic representation learning.