# About exchanging expectation and supremum for conditional Wasserstein GANs

arXiv: https://arxiv.org/abs/2103.13906 · HTML: https://mlsanity.com/paper/2103.13906/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 12 references that are in the map.
- First posted on arXiv 25 Mar 2021.

## Builds on (most cited first, 7 of 7)

- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [Banach Wasserstein GAN](https://mlsanity.com/paper/1806.06621/) (2018)
- [High Perceptual Quality Image Denoising with a Posterior Sampling CGAN](https://mlsanity.com/paper/2103.04192/) (2021)
- [Single Image Haze Removal Using Conditional Wasserstein Generative Adversarial Networks](https://mlsanity.com/paper/1903.00395/) (2019)

## Built on by (most cited first, 3 of 3)

- [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024)
- [Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances](https://mlsanity.com/paper/2310.13433/) (2023)
- [Extended Wasserstein-GAN Approach to Causal Distribution Learning: Density-Free Estimation and Minimax Optimality](https://mlsanity.com/paper/2605.10206/) (2026)

## Abstract (arXiv)

In cases where a Wasserstein GAN depends on a condition the latter is usually handled via an expectation within the loss function. Depending on the way this is motivated, the discriminator is either required to be Lipschitz-1 in both or in only one of its arguments. For the weaker requirement to become usable one needs to exchange a supremum and an expectation. This is a mathematically perilous operation, which is, so far, only partially justified in the literature. This short mathematical note intends to fill this gap and provides the mathematical rationale for discriminators that are only partially Lipschitz-1 for cases where this approach is more appropriate or successful.