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The manifold hypothesis states that many kinds of high-dimensional data areconcentrated near a low-dimensional manifold. If the topology of this datamanifold is non-trivial, a continuous encoder network cannot embed it in aone-to-one manner without creating holes of low density in the latent space.This is at odds with the Gaussian prior assumption typically made inVariational Auto-Encoders (VAEs), because the density of a Gaussianconcentrates 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 differentiablesymmetry groups (Lie groups), such as the group of 3D rotations$\operatorname{SO}(3)$. We show how a VAE with $\operatorname{SO}(3)$-valuedlatent variables can be constructed, by extending the reparameterization trickto compact connected Lie groups. Our experiments show that choosingmanifold-valued latent variables that match the topology of the latent datamanifold, is crucial to preserve the topological structure and learn awell-behaved latent space.

Explorations in Homeomorphic Variational Auto-Encoding