A standard Variational Autoencoder, with a Euclidean latent space, is
structurally incapable of capturing topological properties of certain datasets.
To remove topological obstructions, we introduce Diffusion Variational
Autoencoders with arbitrary manifolds as a latent space. A Diffusion
Variational Autoencoder uses transition kernels of Brownian motion on the
manifold. In particular, it uses properties of the Brownian motion to implement
the reparametrization trick and fast approximations to the KL divergence. We
show that the Diffusion Variational Autoencoder is capable of capturing
topological properties of synthetic datasets. Additionally, we train MNIST on
spheres, tori, projective spaces, SO(3), and a torus embedded in R3. Although a
natural dataset like MNIST does not have latent variables with a clear-cut
topological structure, training it on a manifold can still highlight
topological and geometrical properties.

本文介绍了使用扩散变分自编码器作为潜在空间的任意流形来解决标准变分自编码器无法捕捉某些数据集拓扑性质的问题。我们证明了其可以捕捉合成数据集的拓扑性质，并在各种流形上对 MNIST 数据集进行了训练。