Diffusion Variational Autoencoders

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.

10 pages, 8 figures Added an appendix with derivation of asymptotic expansion of KL divergence for heat kernel on arbitrary Riemannian manifolds, and an appendix with new experiments on binarized MNIST. Added a previously missing factor in the asymptotic expansion of the heat kernel and corrected a coefficient in asymptotic expansion KL divergence; further minor edits

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Cited by in corpus (9)

• Continuous Hierarchical Representations with Poincaré Variational Auto-Encoders
• Variational Autoencoders with Riemannian Brownian Motion Priors
• Geometry-Aware Hamiltonian Variational Auto-Encoder
• Variational Autoencoder with Learned Latent Structure
• Increasing Expressivity of a Hyperspherical VAE
• Disentanglement with Hyperspherical Latent Spaces using Diffusion Variational Autoencoders
• A Metric Space for Point Process Excitations
• Quantifying and Learning Linear Symmetry-Based Disentanglement
• Continuous normalizing flows on manifolds