  Normalizing flows and variational autoencoders are powerful generative models
that can represent complicated density functions. However, they both impose
constraints on the models: Normalizing flows use bijective transformations to
model densities whereas VAEs learn stochastic transformations that are
non-invertible and thus typically do not provide tractable estimates of the
marginal likelihood. In this paper, we introduce SurVAE Flows: A modular
framework of composable transformations that encompasses VAEs and normalizing
flows. SurVAE Flows bridge the gap between normalizing flows and VAEs with
surjective transformations, wherein the transformations are deterministic in
one direction -- thereby allowing exact likelihood computation, and stochastic
in the reverse direction -- hence providing a lower bound on the corresponding
likelihood. We show that several recently proposed methods, including
dequantization and augmented normalizing flows, can be expressed as SurVAE
Flows. Finally, we introduce common operations such as the max value, the
absolute value, sorting and stochastic permutation as composable layers in
SurVAE Flows.
