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.

翻译：我们提出一种高斯流形变分自编码器（GM-VAE），其隐空间由一组高斯分布构成。已知以Fisher信息度量定义的单变量高斯分布集合构成双曲空间，我们称之为高斯流形。为学习基于高斯流形的VAE，我们提出基于Kullback-Leibler散度的伪高斯流形正态分布（即平方Fisher-Rao距离的局部近似），用以定义隐空间上的密度函数。实验表明，GM-VAE在两项不同任务中展现出有效性：图像数据集的密度估计和基于模型的强化学习中的环境建模。在密度估计任务中，GM-VAE优于其他双曲型和欧几里得型VAE变体，并在基于模型的强化学习中表现出竞争性能。我们观察到该模型具备强数值稳定性，有效解决了以往双曲型VAE中常见的局限性问题。