VAEs and Flows are two of the most popular methods for density estimation (well, except GANs I guess, but nevermind... 😱). In this work we will argue they are really two sides of the same coin. A flow is based on deterministically transforming an input density through an invertible transformation to a target density. If the transformation changes a volume element we pick up a log-Jacobian term. After decomposing the ELBO in the only way that was not yet considered in the literature, we find that the log-Jacobian corresponds to log[p(x|z)/q(z|x)] of a VAE, where the maps q and p are now stochastic. This suggests a third possibility that bridges the gap between the two: a surjective map which is deterministic and surjective in one direction, and probabilistic in the reverse direction. We find that these ideas unify many methods out there in the literature, such as dequantization, and augmented flows, and we also add a few new methods of our own based on our SurVAE Flows framework. If time permits I will also say a few words on a new type of flow based on the exponential map which is trivially invertible and adds a new tool to the invertible flows toolbox.

Joint work with Didrik Nielsen, Priyank Jaini, Emiel Hoogeboom.