  We propose a Gaussian manifold variational auto-encoder (GM-VAE) whose latent
space consists of a set of Gaussian distributions. It is known that the set of
the univariate Gaussian distributions with the Fisher information metric form a
hyperbolic space, which we call a Gaussian manifold. To learn the VAE endowed
with the Gaussian manifolds, we propose a pseudo-Gaussian manifold normal
distribution based on the Kullback-Leibler divergence, a local approximation of
the squared Fisher-Rao distance, to define a density over the latent space. In
experiments, we demonstrate the efficacy of GM-VAE on two different tasks:
density estimation of image datasets and environment modeling in model-based
reinforcement learning. GM-VAE outperforms the other variants of hyperbolic-
and Euclidean-VAEs on density estimation tasks and shows competitive
performance in model-based reinforcement learning. We observe that our model
provides strong numerical stability, addressing a common limitation reported in
previous hyperbolic-VAEs.
