Mixed-curvature Variational Autoencoders
Authors: Ondrej Skopek, Octavian-Eugen Ganea, Gary Bécigneul
Keywords: variational autoencoders, riemannian manifolds, non-Euclidean geometry
TL;DR: Variational Autoencoders with latent spaces modeled as products of constant curvature Riemannian manifolds improve on image reconstruction over single-manifold variants.
Abstract: Euclidean space has historically been the typical workhorse geometry for machine learning applications due to its power and simplicity. However, it has recently been shown that geometric spaces with constant non-zero curvature improve representations and performance on a variety of data types and downstream tasks. Consequently, generative models like Variational Autoencoders (VAEs) have been successfully generalized to elliptical and hyperbolic latent spaces. While these approaches work well on data with particular kinds of biases e.g. tree-like data for a hyperbolic VAE, there exists no generic approach unifying and leveraging all three models. We develop a Mixed-curvature Variational Autoencoder, an efficient way to train a VAE whose latent space is a product of constant curvature Riemannian manifolds, where the per-component curvature is fixed or learnable. This generalizes the Euclidean VAE to curved latent spaces and recovers it when curvatures of all latent space components go to 0.