# Generative Modeling by Minimizing the Wasserstein-2 Loss

arXiv: https://arxiv.org/abs/2406.13619 · HTML: https://mlsanity.com/paper/2406.13619/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 26 references that are in the map.
- First posted on arXiv 19 Jun 2024.

## Builds on (most cited first, 12 of 12)

- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [NIPS 2016 Tutorial: Generative Adversarial Networks](https://mlsanity.com/paper/1701.00160/) (2016)
- [Unrolled Generative Adversarial Networks](https://mlsanity.com/paper/1611.02163/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [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)
- [Adversarial Computation of Optimal Transport Maps](https://mlsanity.com/paper/1906.09691/) (2019)
- [Hierarchical Optimal Transport for Unsupervised Domain Adaptation](https://mlsanity.com/paper/2112.02073/) (2021)
- [GANs as Gradient Flows that Converge](https://mlsanity.com/paper/2205.02910/) (2022)

## Built on by (most cited first, 3 of 3)

- [Linear convergence of proximal descent schemes on the Wasserstein space](https://mlsanity.com/paper/2411.15067/) (2024)
- [Mean-Field Langevin Diffusions with Density-dependent Temperature](https://mlsanity.com/paper/2507.20958/) (2025)
- [On optimal solutions of classical and sliced Wasserstein GANs with non-Gaussian data](https://mlsanity.com/paper/2509.06505/) (2025)

## Abstract (arXiv)

This paper develops a generative model by minimizing the second-order Wasserstein loss (the W2 loss) through a distribution-dependent ordinary differential equation (ODE), whose dynamics involves the Kantorovich potential associated with the true data distribution and a current estimate of it. A main result shows that the time-marginal laws of the ODE form a gradient flow for the W2 loss, which converges exponentially to the true data distribution. An Euler scheme for the ODE is proposed and it is shown to recover the gradient flow for the W2 loss in the limit. An algorithm is designed by following the scheme and applying persistent training, which naturally fits our gradient-flow approach. In both low- and high-dimensional experiments, our algorithm outperforms Wasserstein generative adversarial networks by increasing the level of persistent training appropriately.