# Wasserstein-2 Generative Networks

arXiv: https://arxiv.org/abs/1909.13082 · HTML: https://mlsanity.com/paper/1909.13082/

## Key facts

- Cited by 97 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2019.

## Built on by (most cited first, 30 of 97)

- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [ReinFlow: Fine-tuning Flow Matching Policy with Online Reinforcement Learning](https://mlsanity.com/paper/2505.22094/) (2025)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Score-based Generative Neural Networks for Large-Scale Optimal Transport](https://mlsanity.com/paper/2110.03237/) (2021)
- [Variational Wasserstein gradient flow](https://mlsanity.com/paper/2112.02424/) (2021)
- [Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization](https://mlsanity.com/paper/2102.01752/) (2021)
- [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)
- [Neural Monge Map estimation and its applications](https://mlsanity.com/paper/2106.03812/) (2021)
- [Normalizing flow neural networks by JKO scheme](https://mlsanity.com/paper/2212.14424/) (2022)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [Debias Coarsely, Sample Conditionally: Statistical Downscaling through Optimal Transport and Probabilistic Diffusion Models](https://mlsanity.com/paper/2305.15618/) (2023)
- [Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translation](https://mlsanity.com/paper/2311.15100/) (2023)
- [The Monge Gap: A Regularizer to Learn All Transport Maps](https://mlsanity.com/paper/2302.04953/) (2023)
- [Kernel Neural Optimal Transport](https://mlsanity.com/paper/2205.15269/) (2022)
- [On amortizing convex conjugates for optimal transport](https://mlsanity.com/paper/2210.12153/) (2022)
- [Meta Optimal Transport](https://mlsanity.com/paper/2206.05262/) (2022)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Towards Explaining Distribution Shifts](https://mlsanity.com/paper/2210.10275/) (2022)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [Efficient Gradient Flows in Sliced-Wasserstein Space](https://mlsanity.com/paper/2110.10972/) (2021)
- [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)

## Abstract (arXiv)

We propose a novel end-to-end non-minimax algorithm for training optimal transport mappings for the quadratic cost (Wasserstein-2 distance). The algorithm uses input convex neural networks and a cycle-consistency regularization to approximate Wasserstein-2 distance. In contrast to popular entropic and quadratic regularizers, cycle-consistency does not introduce bias and scales well to high dimensions. From the theoretical side, we estimate the properties of the generative mapping fitted by our algorithm. From the practical side, we evaluate our algorithm on a wide range of tasks: image-to-image color transfer, latent space optimal transport, image-to-image style transfer, and domain adaptation.