  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.
