The manifold hypothesis states that many kinds of high-dimensional data are
concentrated near a low-dimensional manifold. If the topology of this data
manifold is non-trivial, a continuous encoder network cannot embed it in a
one-to-one manner without creating holes of low density in the latent space.
This is at odds with the Gaussian prior assumption typically made in
Variational Auto-Encoders (VAEs), because the density of a Gaussian
concentrates near a blob-like manifold.
In this paper we investigate the use of manifold-valued latent variables.
Specifically, we focus on the important case of continuously differentiable
symmetry groups (Lie groups), such as the group of 3D rotations
$\operatorname{SO}(3)$. We show how a VAE with $\operatorname{SO}(3)$-valued
latent variables can be constructed, by extending the reparameterization trick
to compact connected Lie groups. Our experiments show that choosing
manifold-valued latent variables that match the topology of the latent data
manifold, is crucial to preserve the topological structure and learn a
well-behaved latent space.

本文研究了流形值潜变量的使用，特别是连续可微的对称群（李群）的情况，展示了如何通过将再参数化技巧扩展到紧连李群来构造带有李群潜变量的变分自编码器，并通过实验展示了匹配潜在数据流形拓扑结构的流形值潜变量对于保留拓扑结构和学习良好的潜空间至关重要。