# Diffusion Variational Autoencoders

## TL;DR

This paper introduces Diffusion Variational Autoencoders that utilize arbitrary manifolds as latent spaces, enabling the capture of topological properties in datasets, which standard VAEs cannot achieve.

## Contribution

It proposes a novel VAE framework using Brownian motion on manifolds, allowing for topological and geometric data representation.

## Key findings

- Successfully captures topological features of synthetic datasets
- Demonstrates training on MNIST with various manifolds revealing topological properties
- Uses Brownian motion properties for reparametrization and KL divergence approximation

## Abstract

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.