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.

翻译：鉴于球的半径量的指数增长,双曲空间能够将树嵌入任意的小扭曲,从而引起代表等级数据集的广泛关注。然而,这种指数增长属性是以数字不稳定的价格产生的,因此培训双曲学习模型有时会导致灾难性的NNN问题,在浮动点算术中遇到无法反映的数值。在这项工作中,我们仔细分析双曲空间的两个流行模型,即Poincar\'e球和Lorentz模型的局限性。我们首先显示,在64位数算术系统下,Poincar\'e球比Lorentz模型的正确代表点能力要大得多。然后,我们从优化的角度从理论上验证Lorentz模型优于Poincar\'e球的优势。鉴于两种模型的数值限制,我们找出了一种超曲解的超曲空间的极美化,可以减轻这些限制。我们进一步将这种Euclidean paremterizizizizization 扩大到了超低调的超高压平板。