  Since the introduction of Generative Adversarial Networks (GANs) and
Variational Autoencoders (VAE), the literature on generative modelling has
witnessed an overwhelming resurgence. The impressive, yet elusive empirical
performance of GANs has lead to the rise of many GAN-VAE hybrids, with the
hopes of GAN level performance and additional benefits of VAE, such as an
encoder for feature reduction, which is not offered by GANs. Recently, the
Wasserstein Autoencoder (WAE) was proposed, achieving performance similar to
that of GANs, yet it is still unclear whether the two are fundamentally
different or can be further improved into a unified model. In this work, we
study the $f$-GAN and WAE models and make two main discoveries. First, we find
that the $f$-GAN and WAE objectives partake in a primal-dual relationship and
are equivalent under some assumptions, which then allows us to explicate the
success of WAE. Second, the equivalence result allows us to, for the first
time, prove generalization bounds for Autoencoder models, which is a pertinent
problem when it comes to theoretical analyses of generative models.
Furthermore, we show that the WAE objective is related to other statistical
quantities such as the $f$-divergence and in particular, upper bounded by the
Wasserstein distance, which then allows us to tap into existing efficient
(regularized) optimal transport solvers. Our findings thus present the first
primal-dual relationship between GANs and Autoencoder models, comment on
generalization abilities and make a step towards unifying these models.
