# Wasserstein GANs with Gradient Penalty Compute Congested Transport

arXiv: https://arxiv.org/abs/2109.00528 · HTML: https://mlsanity.com/paper/2109.00528/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 26 references that are in the map.
- First posted on arXiv 1 Sep 2021.

## Builds on (most cited first, 10 of 10)

- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [On the regularization of Wasserstein GANs](https://mlsanity.com/paper/1709.08894/) (2017)
- [Wasserstein GANs Work Because They Fail (to Approximate the Wasserstein Distance)](https://mlsanity.com/paper/2103.01678/) (2021)
- [Some Theoretical Insights into Wasserstein GANs](https://mlsanity.com/paper/2006.02682/) (2020)
- [How Well Do WGANs Estimate the Wasserstein Metric?](https://mlsanity.com/paper/1910.03875/) (2019)
- [On the estimation of the Wasserstein distance in generative models](https://mlsanity.com/paper/1910.00888/) (2019)
- [Primal-Dual Wasserstein GAN](https://mlsanity.com/paper/1805.09575/) (2018)

## Built on by (most cited first, 6 of 6)

- [Generative Adversarial Neural Operators](https://mlsanity.com/paper/2205.03017/) (2022)
- [Kantorovich Strikes Back! Wasserstein GANs are not Optimal Transport?](https://mlsanity.com/paper/2206.07767/) (2022)
- [A new method for determining Wasserstein 1 optimal transport maps from Kantorovich potentials, with deep learning applications](https://mlsanity.com/paper/2211.00820/) (2022)
- [Trust the Critics: Generatorless and Multipurpose WGANs with Initial Convergence Guarantees](https://mlsanity.com/paper/2111.15099/) (2021)
- [High Accuracy and Cost-Saving Active Learning 3D WD-UNet for Airway Segmentation](https://mlsanity.com/paper/2310.05638/) (2023)
- [PuckTrick: A Library for Making Synthetic Data More Realistic](https://mlsanity.com/paper/2506.18499/) (2025)

## Abstract (arXiv)

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.