# Adversarial Computation of Optimal Transport Maps

arXiv: https://arxiv.org/abs/1906.09691 · HTML: https://mlsanity.com/paper/1906.09691/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jun 2019.

## Built on by (most cited first, 12 of 12)

- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Score-based Generative Neural Networks for Large-Scale Optimal Transport](https://mlsanity.com/paper/2110.03237/) (2021)
- [Improving Mini-batch Optimal Transport via Partial Transportation](https://mlsanity.com/paper/2108.09645/) (2021)
- [On amortizing convex conjugates for optimal transport](https://mlsanity.com/paper/2210.12153/) (2022)
- [On Transportation of Mini-batches: A Hierarchical Approach](https://mlsanity.com/paper/2102.05912/) (2021)
- [ENOT: Expectile Regularization for Fast and Accurate Training of Neural Optimal Transport](https://mlsanity.com/paper/2403.03777/) (2024)
- [GAN Estimation of Lipschitz Optimal Transport Maps](https://mlsanity.com/paper/2202.07965/) (2022)
- [Generative Modeling by Minimizing the Wasserstein-2 Loss](https://mlsanity.com/paper/2406.13619/) (2024)
- [Continual Learning of Generative Models with Limited Data: From Wasserstein-1 Barycenter to Adaptive Coalescence](https://mlsanity.com/paper/2101.09225/) (2021)
- [A Differential Equation Approach for Wasserstein GANs and Beyond](https://mlsanity.com/paper/2405.16351/) (2024)
- [Neural solver for Wasserstein Geodesics and optimal transport dynamics](https://mlsanity.com/paper/2602.22003/) (2026)

## Abstract (arXiv)

Computing optimal transport maps between high-dimensional and continuous distributions is a challenging problem in optimal transport (OT). Generative adversarial networks (GANs) are powerful generative models which have been successfully applied to learn maps across high-dimensional domains. However, little is known about the nature of the map learned with a GAN objective. To address this problem, we propose a generative adversarial model in which the discriminator's objective is the 2-Wasserstein metric. We show that during training, our generator follows the W2-geodesic between the initial and the target distributions. As a consequence, it reproduces an optimal map at the end of training. We validate our approach empirically in both low-dimensional and high-dimensional continuous settings, and show that it outperforms prior methods on image data.