  Euclidean geometry has historically been the typical "workhorse" 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.
