Implicit Bias in Hyperbolic Multiclass Learning Is Governed by Busemann Risk: A Drift Coefficient Decides Whether Embeddings Escape to the Ideal Boundary or Return Inward
Studying the implicit bias of Riemannian gradient flow for hyperbolic multiclass classification with fixed prototypes, this work decomposes large-radius distances into a radial term and a Busemann direction term, proving a radial dichotomy in which the sign of a drift coefficient decides whether the radius escapes to the ideal boundary or returns inward, and showing that boundary directions converge to critical points of the Busemann risk, with an asymptotic classifier and decision boundaries on the ideal boundary.