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

arXiv: https://arxiv.org/abs/2610.07131 · HTML: https://mlsanity.com/paper/2610.07131/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 51 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Builds on (most cited first, 11 of 11)

- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Hyperspherical Prototype Networks](https://mlsanity.com/paper/1901.10514/) (2019)
- [Risk and parameter convergence of logistic regression](https://mlsanity.com/paper/1803.07300/) (2018)
- [Neural Networks on Symmetric Spaces of Noncompact Type](https://mlsanity.com/paper/2601.01097/) (2026)
- [Hyperbolic Busemann Neural Networks](https://mlsanity.com/paper/2602.18858/) (2026)
- [Hyperbolic Neural Networks++](https://mlsanity.com/paper/2006.08210/) (2020)

## Abstract (arXiv)

We study the implicit bias of Riemannian gradient flow for hyperbolic multiclass classification with fixed class prototypes in hyperbolic space mathbbHn. 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)=frac12log 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 partialmathbbHn. These results provide a rigorous asymptotic perspective on two phenomena we refer to as boundary saturation and near-boundary clustering in hyperbolic representation learning.