The Numerical Stability of Hyperbolic Representation Learning

The hyperbolic space is widely used for representing hierarchical datasets due to its ability to embed trees with small distortion. However, this 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 analyze the limitations of two popular models for the hyperbolic space, namely, the Poincaré ball and the Lorentz model. We find that, under the 64-bit arithmetic system, the Poincaré ball has a relatively larger capacity than the Lorentz model for correctly representing points. However, the Lorentz model is superior to the Poincaré ball from the perspective of optimization, which we theoretically validate. To address these limitations, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these issues. We further extend this Euclidean parametrization to hyperbolic hyperplanes and demonstrate its effectiveness in improving the performance of hyperbolic SVM.