# 2-Wasserstein Approximation via Restricted Convex Potentials with Application to Improved Training for GANs

arXiv: https://arxiv.org/abs/1902.07197 · HTML: https://mlsanity.com/paper/1902.07197/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Feb 2019.

## Built on by (most cited first, 30 of 33)

- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [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)
- [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)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [Light Schr"odinger Bridge](https://mlsanity.com/paper/2310.01174/) (2023)
- [Reversible Gromov-Monge Sampler for Simulation-Based Inference](https://mlsanity.com/paper/2109.14090/) (2021)
- [Light Unbalanced Optimal Transport](https://mlsanity.com/paper/2303.07988/) (2023)
- [Computing high-dimensional optimal transport by flow neural networks](https://mlsanity.com/paper/2305.11857/) (2023)
- [ENOT: Expectile Regularization for Fast and Accurate Training of Neural Optimal Transport](https://mlsanity.com/paper/2403.03777/) (2024)
- [Neural approximation of Wasserstein distance via a universal architecture for symmetric and factorwise group invariant functions](https://mlsanity.com/paper/2308.00273/) (2023)
- [Monotone Generative Modeling via a Gromov-Monge Embedding](https://mlsanity.com/paper/2311.01375/) (2023)
- [Physics Informed Convex Artificial Neural Networks (PICANNs) for Optimal Transport based Density Estimation](https://mlsanity.com/paper/2104.01194/) (2021)
- [Differentiable Cost-Parameterized Monge Map Estimators](https://mlsanity.com/paper/2406.08399/) (2024)
- [Deep Generative Quantile Bayes](https://mlsanity.com/paper/2410.08378/) (2024)
- [A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers](https://mlsanity.com/paper/2502.01310/) (2025)
- [Neural Optimal Transport Meets Multivariate Conformal Prediction](https://mlsanity.com/paper/2509.25444/) (2025)
- [Hyper Input Convex Neural Networks for Shape Constrained Learning and Optimal Transport](https://mlsanity.com/paper/2604.26942/) (2026)
- [Displacement-Sparse Neural Optimal Transport](https://mlsanity.com/paper/2502.01889/) (2025)
- [Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport](https://mlsanity.com/paper/2510.01153/) (2025)
- [Deep Legendre Transform](https://mlsanity.com/paper/2512.19649/) (2025)
- [Connecting adversarial attacks and optimal transport for domain adaptation](https://mlsanity.com/paper/2205.15424/) (2022)
- [Distribution estimation and change-point estimation for time series via DNN-based GANs](https://mlsanity.com/paper/2211.14577/) (2022)
- [Hyperparameter Trajectory Inference with Conditional Lagrangian Optimal Transport](https://mlsanity.com/paper/2603.01771/) (2026)

## Abstract (arXiv)

We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optimal potential function is convex and the optimal transport map is the gradient of the optimal potential function. Using this geometric structure, we restrict the optimization problem to different parametrized classes of convex functions and pay special attention to the class of input-convex neural networks. We analyze the statistical generalization and the discriminative power of the resulting approximate metric, and we prove a restricted moment-matching property for the approximate optimal map. Finally, we discuss a numerical algorithm to solve the restricted optimization problem and provide numerical experiments to illustrate and compare the proposed approach with the established regularization-based approaches. We further discuss practical implications of our proposal in a modular and interpretable design for GANs which connects the generator training with discriminator computations to allow for learning an overall composite generator.