# How Well Do WGANs Estimate the Wasserstein Metric?

arXiv: https://arxiv.org/abs/1910.03875 · HTML: https://mlsanity.com/paper/1910.03875/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2019.

## Built on by (most cited first, 17 of 17)

- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Improving Mini-batch Optimal Transport via Partial Transportation](https://mlsanity.com/paper/2108.09645/) (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)
- [On Transportation of Mini-batches: A Hierarchical Approach](https://mlsanity.com/paper/2102.05912/) (2021)
- [Adversarial Score Distillation: When score distillation meets GAN](https://mlsanity.com/paper/2312.00739/) (2023)
- [Wasserstein GANs with Gradient Penalty Compute Congested Transport](https://mlsanity.com/paper/2109.00528/) (2021)
- [SDYN-GANs: Adversarial Learning Methods for Multistep Generative Models for General Order Stochastic Dynamics](https://mlsanity.com/paper/2302.03663/) (2023)
- [Convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings](https://mlsanity.com/paper/2101.01429/) (2021)
- [Training Wasserstein GANs without gradient penalties](https://mlsanity.com/paper/2110.14150/) (2021)
- [Approximation Theory, Computing, and Deep Learning on the Wasserstein Space](https://mlsanity.com/paper/2310.19548/) (2023)
- [Adversarial Reweighting Guided by Wasserstein Distance for Bias Mitigation](https://mlsanity.com/paper/2311.12684/) (2023)
- [Variational Neural Stochastic Differential Equations with Change Points](https://mlsanity.com/paper/2411.00635/) (2024)
- [Learning to Emulate Chaos: Adversarial Optimal Transport Regularization](https://mlsanity.com/paper/2604.21097/) (2026)
- [A method for quantifying the generalization capabilities of generative models for solving Ising models](https://mlsanity.com/paper/2405.03435/) (2024)
- [Multivariate Distributional Reinforcement Learning Using Sliced Divergences](https://mlsanity.com/paper/2605.31222/) (2026)

## Abstract (arXiv)

Generative modelling is often cast as minimizing a similarity measure between a data distribution and a model distribution. Recently, a popular choice for the similarity measure has been the Wasserstein metric, which can be expressed in the Kantorovich duality formulation as the optimum difference of the expected values of a potential function under the real data distribution and the model hypothesis. In practice, the potential is approximated with a neural network and is called the discriminator. Duality constraints on the function class of the discriminator are enforced approximately, and the expectations are estimated from samples. This gives at least three sources of errors: the approximated discriminator and constraints, the estimation of the expectation value, and the optimization required to find the optimal potential. In this work, we study how well the methods, that are used in generative adversarial networks to approximate the Wasserstein metric, perform. We consider, in particular, the c-transform formulation, which eliminates the need to enforce the constraints explicitly. We demonstrate that the c-transform allows for a more accurate estimation of the true Wasserstein metric from samples, but surprisingly, does not perform the best in the generative setting.