Hyperbolic Busemann Learning with Ideal Prototypes

TL;DR

This paper introduces Hyperbolic Busemann Learning, positioning prototypes on the ideal boundary of hyperbolic space without label priors, supported by theory and empirical results showing improved classification performance.

Contribution

It proposes a novel hyperbolic prototype placement method on the ideal boundary, eliminating the need for label priors and providing theoretical and empirical validation.

Findings

Outperforms recent hyperbolic prototype methods

Provides a natural interpretation of classification confidence

Theoretically equivalent to logistic regression in 1D

Abstract

Hyperbolic space has become a popular choice of manifold for representation learning of various datatypes from tree-like structures and text to graphs. Building on the success of deep learning with prototypes in Euclidean and hyperspherical spaces, a few recent works have proposed hyperbolic prototypes for classification. Such approaches enable effective learning in low-dimensional output spaces and can exploit hierarchical relations amongst classes, but require privileged information about class labels to position the hyperbolic prototypes. In this work, we propose Hyperbolic Busemann Learning. The main idea behind our approach is to position prototypes on the ideal boundary of the Poincar\'e ball, which does not require prior label knowledge. To be able to compute proximities to ideal prototypes, we introduce the penalised Busemann loss. We provide theory supporting the use of ideal prototypes and the proposed loss by proving its equivalence to logistic regression in the one-dimensional case. Empirically, we show that our approach provides a natural interpretation of classification confidence, while outperforming recent hyperspherical and hyperbolic prototype approaches.