  Wasserstein GANs with Gradient Penalty (WGAN-GP) are a very popular method
for training generative models to produce high quality synthetic data. While
WGAN-GP were initially developed to calculate the Wasserstein 1 distance
between generated and real data, recent works (e.g. [23]) have provided
empirical evidence that this does not occur, and have argued that WGAN-GP
perform well not in spite of this issue, but because of it. In this paper we
show for the first time that WGAN-GP compute the minimum of a different optimal
transport problem, the so-called congested transport [7]. Congested transport
determines the cost of moving one distribution to another under a transport
model that penalizes congestion. For WGAN-GP, we find that the congestion
penalty has a spatially varying component determined by the sampling strategy
used in [12] which acts like a local speed limit, making congestion cost less
in some regions than others. This aspect of the congested transport problem is
new, in that the congestion penalty turns out to be unbounded and depends on
the distributions to be transported, and so we provide the necessary
mathematical proofs for this setting. One facet of our discovery is a formula
connecting the gradient of solutions to the optimization problem in WGAN-GP to
the time averaged momentum of the optimal mass flow. This is in contrast to the
gradient of Kantorovich potentials for the Wasserstein 1 distance, which is
just the normalized direction of flow. Based on this and other considerations,
we speculate on how our results explain the observed performance of WGAN-GP.
Beyond applications to GANs, our theorems also point to the possibility of
approximately solving large scale congested transport problems using neural
network techniques.
