The Numerical Stability of Hyperbolic Representation Learning

TL;DR

This paper analyzes the numerical stability issues in hyperbolic space models for hierarchical data embedding, compares popular models, and proposes a Euclidean parametrization to improve stability and performance.

Contribution

It provides a theoretical comparison of Poincaré ball and Lorentz models, and introduces a Euclidean parametrization to mitigate numerical instability in hyperbolic learning.

Findings

Lorentz model has better capacity than Poincaré ball under 64-bit arithmetic.

The Euclidean parametrization alleviates numerical limitations.

Improved hyperbolic SVM performance with the new parametrization.

Abstract

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.