# Optimal 1-Wasserstein Distance for WGANs

arXiv: https://arxiv.org/abs/2201.02824 · HTML: https://mlsanity.com/paper/2201.02824/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 46 references that are in the map.
- First posted on arXiv 8 Jan 2022.

## Builds on (most cited first, 20 of 20)

- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [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)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [Generating Videos with Scene Dynamics](https://mlsanity.com/paper/1609.02612/) (2016)
- [SeqGAN: Sequence Generative Adversarial Nets with Policy Gradient](https://mlsanity.com/paper/1609.05473/) (2016)
- [Are GANs Created Equal? A Large-Scale Study](https://mlsanity.com/paper/1711.10337/) (2017)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Pros and Cons of GAN Evaluation Measures](https://mlsanity.com/paper/1802.03446/) (2018)
- [How Well Generative Adversarial Networks Learn Distributions](https://mlsanity.com/paper/1811.03179/) (2018)
- [Towards GAN Benchmarks Which Require Generalization](https://mlsanity.com/paper/2001.03653/) (2020)
- [Some Theoretical Insights into Wasserstein GANs](https://mlsanity.com/paper/2006.02682/) (2020)
- [Statistical guarantees for generative models without domination](https://mlsanity.com/paper/2010.09237/) (2020)
- [Nonparametric Density Estimation under Adversarial Losses](https://mlsanity.com/paper/1805.08836/) (2018)
- [Learning disconnected manifolds: a no GANs land](https://mlsanity.com/paper/2006.04596/) (2020)
- [Lipschitz Generative Adversarial Nets](https://mlsanity.com/paper/1902.05687/) (2019)
- [Nonparametric Density Estimation & Convergence Rates for GANs under Besov IPM Losses](https://mlsanity.com/paper/1902.03511/) (2019)
- [Generalization Properties of Optimal Transport GANs with Latent Distribution Learning](https://mlsanity.com/paper/2007.14641/) (2020)
- [Reversible Gromov-Monge Sampler for Simulation-Based Inference](https://mlsanity.com/paper/2109.14090/) (2021)

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

- [Can Push-forward Generative Models Fit Multimodal Distributions?](https://mlsanity.com/paper/2206.14476/) (2022)
- [Statistical guarantees for denoising reflected diffusion models](https://mlsanity.com/paper/2411.01563/) (2024)
- [Active Learning for Regression based on Wasserstein distance and GroupSort Neural Networks](https://mlsanity.com/paper/2403.15108/) (2024)

## Abstract (arXiv)

The mathematical forces at work behind Generative Adversarial Networks raise challenging theoretical issues. Motivated by the important question of characterizing the geometrical properties of the generated distributions, we provide a thorough analysis of Wasserstein GANs (WGANs) in both the finite sample and asymptotic regimes. We study the specific case where the latent space is univariate and derive results valid regardless of the dimension of the output space. We show in particular that for a fixed sample size, the optimal WGANs are closely linked with connected paths minimizing the sum of the squared Euclidean distances between the sample points. We also highlight the fact that WGANs are able to approach (for the 1-Wasserstein distance) the target distribution as the sample size tends to infinity, at a given convergence rate and provided the family of generative Lipschitz functions grows appropriately. We derive in passing new results on optimal transport theory in the semi-discrete setting.