# (q,p)-Wasserstein GANs: Comparing Ground Metrics for Wasserstein GANs

arXiv: https://arxiv.org/abs/1902.03642 · HTML: https://mlsanity.com/paper/1902.03642/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2019.

## Built on by (most cited first, 10 of 10)

- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Wasserstein GANs Work Because They Fail (to Approximate the Wasserstein Distance)](https://mlsanity.com/paper/2103.01678/) (2021)
- [Kantorovich Strikes Back! Wasserstein GANs are not Optimal Transport?](https://mlsanity.com/paper/2206.07767/) (2022)
- [An Optimal Transport Perspective on Unpaired Image Super-Resolution](https://mlsanity.com/paper/2202.01116/) (2022)
- [Sample Complexity Bounds for Estimating Probability Divergences under Invariances](https://mlsanity.com/paper/2311.02868/) (2023)
- [ENOT: Expectile Regularization for Fast and Accurate Training of Neural Optimal Transport](https://mlsanity.com/paper/2403.03777/) (2024)
- [Physics Informed Convex Artificial Neural Networks (PICANNs) for Optimal Transport based Density Estimation](https://mlsanity.com/paper/2104.01194/) (2021)
- [Embedding Empirical Distributions for Computing Optimal Transport Maps](https://mlsanity.com/paper/2504.17740/) (2025)
- [Stability of the Monge Map in Semi-Dual Optimal Transport](https://mlsanity.com/paper/2605.05569/) (2026)

## Abstract (arXiv)

Generative Adversial Networks (GANs) have made a major impact in computer vision and machine learning as generative models. Wasserstein GANs (WGANs) brought Optimal Transport (OT) theory into GANs, by minimizing the 1-Wasserstein distance between model and data distributions as their objective function. Since then, WGANs have gained considerable interest due to their stability and theoretical framework. We contribute to the WGAN literature by introducing the family of (q,p)-Wasserstein GANs, which allow the use of more general p-Wasserstein metrics for p≥ 1 in the GAN learning procedure. While the method is able to incorporate any cost function as the ground metric, we focus on studying the lq metrics for q≥ 1. This is a notable generalization as in the WGAN literature the OT distances are commonly based on the l2 ground metric. We demonstrate the effect of different p-Wasserstein distances in two toy examples. Furthermore, we show that the ground metric does make a difference, by comparing different (q,p) pairs on the MNIST and CIFAR-10 datasets. Our experiments demonstrate that changing the ground metric and p can notably improve on the common (q,p) = (2,1) case.