Hyperbolic space is a geometry that is known to be well-suited for representation learning of data with an underlying hierarchical structure. In this paper, we present a novel hyperbolic distribution called \textit{pseudo-hyperbolic Gaussian}, a Gaussian-like distribution on hyperbolic space whose density can be evaluated analytically and differentiated with respect to the parameters. Our distribution enables the gradient-based learning of the probabilistic models on hyperbolic space that could never have been considered before. Also, we can sample from this hyperbolic probability distribution without resorting to auxiliary means like rejection sampling. As applications of our distribution, we develop a hyperbolic-analog of variational autoencoder and a method of probabilistic word embedding on hyperbolic space. We demonstrate the efficacy of our distribution on various datasets including MNIST, Atari 2600 Breakout, and WordNet.

本文提出了新的伪超柏格分布，它是一种基于测地线距离的高斯分布，可用于超半径空间的概率模型。我们展示了该分布在超半径空间上的应用，包括超半径变分自编码器和超半径概率词嵌入。在MNIST、Atari 2600 Breakout和WordNet等多个数据集上验证了该分布的有效性。

基于梯度的超几何空间上的包裹正态分布学习