A Wrapped Normal Distribution on Hyperbolic Space for Gradient-Based Learning

TL;DR

This paper introduces a new hyperbolic distribution called pseudo-hyperbolic Gaussian, enabling gradient-based learning and sampling in hyperbolic space, with applications to hyperbolic autoencoders and word embeddings.

Contribution

It presents the first analytically evaluable and differentiable Gaussian-like distribution on hyperbolic space, facilitating probabilistic modeling and learning.

Findings

Effective on datasets like MNIST, Atari Breakout, and WordNet.

Enables gradient-based training of models on hyperbolic space.

Supports sampling without rejection sampling.

Abstract

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.