Safety
The Implicit Bias of Hyperbolic Representation Learning for Multiclass Data: A Busemann Risk Perspective
arXiv:2610.07131v1 Announce Type: new Abstract: We study the implicit bias of Riemannian gradient flow for hyperbolic multiclass classification with fixed class prototypes in hyperbolic space H^n. Our
arXiv:2610.07131v1 Announce Type: new Abstract: We study the implicit bias of Riemannian gradient flow for hyperbolic multiclass classification with fixed class prototypes in hyperbolic space 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 partialH^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.
Related
- HyCoSeq: Contextual Hyperbolic Representation Learning for Genomic Sequences
- Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective
- Stability Annealing Selects the Implicit Bias of Smoothed Sign Descent: A Rate-Indexed Barrier Path on Separable Data
Source: arXiv cs.LG | 2026-10-07