A Theory of Hyperbolic Prototype Learning

TL;DR

This paper proposes Hyperbolic Prototype Learning, a supervised classification method using hyperbolic space and a novel loss function, with theoretical analysis showing its relation to logistic regression in one dimension.

Contribution

It introduces a new hyperbolic geometry-based learning framework with a novel loss function and explores its theoretical properties, including its equivalence to logistic regression in 1D.

Findings

Hyperbolic Prototype Learning effectively models class prototypes at infinity.

The penalized Busemann loss is a novel, geometry-inspired loss function.

In 1D, the method reduces to logistic regression.

Abstract

We introduce Hyperbolic Prototype Learning, a type of supervised learning, where class labels are represented by ideal points (points at infinity) in hyperbolic space. Learning is achieved by minimizing the 'penalized Busemann loss', a new loss function based on the Busemann function of hyperbolic geometry. We discuss several theoretical features of this setup. In particular, Hyperbolic Prototype Learning becomes equivalent to logistic regression in the one-dimensional case.