A Wrapped Normal Distribution on Hyperbolic Space for Gradient-Based Learning
This work addresses the challenge of probabilistic learning in hyperbolic geometry for hierarchical data representation, enabling new models that were previously infeasible.
The authors tackled the problem of gradient-based probabilistic modeling on hyperbolic space by introducing a novel Gaussian-like distribution that enables analytic density evaluation and differentiation, leading to applications like hyperbolic variational autoencoders and probabilistic word embeddings demonstrated on datasets such as MNIST, Atari 2600 Breakout, and WordNet.
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.