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

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.