# Kantorovich Strikes Back! Wasserstein GANs are not Optimal Transport?

arXiv: https://arxiv.org/abs/2206.07767 · HTML: https://mlsanity.com/paper/2206.07767/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 68 references that are in the map.
- First posted on arXiv 15 Jun 2022.

## Builds on (most cited first, 30 of 36)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Learning with minibatch Wasserstein : asymptotic and gradient properties](https://mlsanity.com/paper/1910.04091/) (2019)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [On the regularization of Wasserstein GANs](https://mlsanity.com/paper/1709.08894/) (2017)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Score-based Generative Neural Networks for Large-Scale Optimal Transport](https://mlsanity.com/paper/2110.03237/) (2021)
- [On the Convergence and Robustness of Training GANs with Regularized Optimal Transport](https://mlsanity.com/paper/1802.08249/) (2018)
- [Improving the Improved Training of Wasserstein GANs: A Consistency Term and Its Dual Effect](https://mlsanity.com/paper/1803.01541/) (2018)
- [Improving Mini-batch Optimal Transport via Partial Transportation](https://mlsanity.com/paper/2108.09645/) (2021)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [Wasserstein GANs Work Because They Fail (to Approximate the Wasserstein Distance)](https://mlsanity.com/paper/2103.01678/) (2021)
- [Kernel Neural Optimal Transport](https://mlsanity.com/paper/2205.15269/) (2022)
- [Discriminator optimal transport](https://mlsanity.com/paper/1910.06832/) (2019)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [GAN and VAE from an Optimal Transport Point of View](https://mlsanity.com/paper/1706.01807/) (2017)
- [How Well Do WGANs Estimate the Wasserstein Metric?](https://mlsanity.com/paper/1910.03875/) (2019)

## Built on by (most cited first, 19 of 19)

- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [Improving Lipschitz-Constrained Neural Networks by Learning Activation Functions](https://mlsanity.com/paper/2210.16222/) (2022)
- [Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmark](https://mlsanity.com/paper/2306.10161/) (2023)
- [Light Schr"odinger Bridge](https://mlsanity.com/paper/2310.01174/) (2023)
- [Extremal Domain Translation with Neural Optimal Transport](https://mlsanity.com/paper/2301.12874/) (2023)
- [Energy-Guided Continuous Entropic Barycenter Estimation for General Costs](https://mlsanity.com/paper/2310.01105/) (2023)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Rethinking Optimal Transport in Offline Reinforcement Learning](https://mlsanity.com/paper/2410.14069/) (2024)
- [Light Unbalanced Optimal Transport](https://mlsanity.com/paper/2303.07988/) (2023)
- [Improving Neural Optimal Transport via Displacement Interpolation](https://mlsanity.com/paper/2410.03783/) (2024)
- [Fast and scalable Wasserstein-1 neural optimal transport solver for single-cell perturbation prediction](https://mlsanity.com/paper/2411.00614/) (2024)
- [Data Interpolants -- That's What Discriminators in Higher-order Gradient-regularized GANs Are](https://mlsanity.com/paper/2306.00785/) (2023)
- [Expert or not? assessing data quality in offline reinforcement learning](https://mlsanity.com/paper/2510.12638/) (2025)
- [Experts-Guided Unbalanced Optimal Transport for ISP Learning from Unpaired and/or Paired Data](https://mlsanity.com/paper/2512.05635/) (2025)
- [Insights into Closed-form IPM-GAN Discriminator Guidance for Diffusion Modeling](https://mlsanity.com/paper/2306.01654/) (2023)
- [Entering the Era of Discrete Diffusion Models: A Benchmark for Schr"odinger Bridges and Entropic Optimal Transport](https://mlsanity.com/paper/2509.23348/) (2025)

## Abstract (arXiv)

Wasserstein Generative Adversarial Networks (WGANs) are the popular generative models built on the theory of Optimal Transport (OT) and the Kantorovich duality. Despite the success of WGANs, it is still unclear how well the underlying OT dual solvers approximate the OT cost (Wasserstein-1 distance, mathbbW1) and the OT gradient needed to update the generator. In this paper, we address these questions. We construct 1-Lipschitz functions and use them to build ray monotone transport plans. This strategy yields pairs of continuous benchmark distributions with the analytically known OT plan, OT cost and OT gradient in high-dimensional spaces such as spaces of images. We thoroughly evaluate popular WGAN dual form solvers (gradient penalty, spectral normalization, entropic regularization, etc.) using these benchmark pairs. Even though these solvers perform well in WGANs, none of them faithfully compute mathbbW1 in high dimensions. Nevertheless, many provide a meaningful approximation of the OT gradient. These observations suggest that these solvers should not be treated as good estimators of mathbbW1, but to some extent they indeed can be used in variational problems requiring the minimization of mathbbW1.