# Lipschitz Generative Adversarial Nets

arXiv: https://arxiv.org/abs/1902.05687 · HTML: https://mlsanity.com/paper/1902.05687/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Feb 2019.

## Built on by (most cited first, 24 of 24)

- [ViTGAN: Training GANs with Vision Transformers](https://mlsanity.com/paper/2107.04589/) (2021)
- [Rebooting ACGAN: Auxiliary Classifier GANs with Stable Training](https://mlsanity.com/paper/2111.01118/) (2021)
- [StudioGAN: A Taxonomy and Benchmark of GANs for Image Synthesis](https://mlsanity.com/paper/2206.09479/) (2022)
- [Some Fundamental Aspects about Lipschitz Continuity of Neural Networks](https://mlsanity.com/paper/2302.10886/) (2023)
- [A Neural Tangent Kernel Perspective of GANs](https://mlsanity.com/paper/2106.05566/) (2021)
- [Quaternion Generative Adversarial Networks](https://mlsanity.com/paper/2104.09630/) (2021)
- [Convolutional Normalization: Improving Deep Convolutional Network Robustness and Training](https://mlsanity.com/paper/2103.00673/) (2021)
- [Decoupled Rationalization with Asymmetric Learning Rates: A Flexible Lipschitz Restraint](https://mlsanity.com/paper/2305.13599/) (2023)
- [Self-Supervised GANs with Label Augmentation](https://mlsanity.com/paper/2106.08601/) (2021)
- [Moreau-Yosida f-divergences](https://mlsanity.com/paper/2102.13416/) (2021)
- [HDRVideo-GAN: Deep Generative HDR Video Reconstruction](https://mlsanity.com/paper/2110.11795/) (2021)
- [Optimal 1-Wasserstein Distance for WGANs](https://mlsanity.com/paper/2201.02824/) (2022)
- [Unveiling the Latent Space Geometry of Push-Forward Generative Models](https://mlsanity.com/paper/2207.10541/) (2022)
- [Quaternion Convolutional Neural Networks: Current Advances and Future Directions](https://mlsanity.com/paper/2307.08663/) (2023)
- [A Wasserstein perspective of Vanilla GANs](https://mlsanity.com/paper/2403.15312/) (2024)
- [Generalization of GANs and overparameterized models under Lipschitz continuity](https://mlsanity.com/paper/2104.02388/) (2021)
- [Fair Supervised Learning with A Simple Random Sampler of Sensitive Attributes](https://mlsanity.com/paper/2311.05866/) (2023)
- [Graph Your Own Prompt](https://mlsanity.com/paper/2509.23373/) (2025)
- [GraN-GAN: Piecewise Gradient Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/2111.03162/) (2021)
- [Stabilizing Multimodal Autoencoders: A Theoretical and Empirical Analysis of Fusion Strategies](https://mlsanity.com/paper/2512.20749/) (2025)
- [Unsupervised Adaptive Semantic Segmentation with Local Lipschitz Constraint](https://mlsanity.com/paper/2105.12939/) (2021)
- [Penalty Gradient Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/2306.13576/) (2023)
- [A New Paradigm for Generative Adversarial Networks based on Randomized Decision Rules](https://mlsanity.com/paper/2306.13641/) (2023)
- [Scaling-based Data Augmentation for Generative Models and its Theoretical Extension](https://mlsanity.com/paper/2410.20780/) (2024)

## Abstract (arXiv)

In this paper, we study the convergence of generative adversarial networks (GANs) from the perspective of the informativeness of the gradient of the optimal discriminative function. We show that GANs without restriction on the discriminative function space commonly suffer from the problem that the gradient produced by the discriminator is uninformative to guide the generator. By contrast, Wasserstein GAN (WGAN), where the discriminative function is restricted to 1-Lipschitz, does not suffer from such a gradient uninformativeness problem. We further show in the paper that the model with a compact dual form of Wasserstein distance, where the Lipschitz condition is relaxed, may also theoretically suffer from this issue. This implies the importance of Lipschitz condition and motivates us to study the general formulation of GANs with Lipschitz constraint, which leads to a new family of GANs that we call Lipschitz GANs (LGANs). We show that LGANs guarantee the existence and uniqueness of the optimal discriminative function as well as the existence of a unique Nash equilibrium. We prove that LGANs are generally capable of eliminating the gradient uninformativeness problem. According to our empirical analysis, LGANs are more stable and generate consistently higher quality samples compared with WGAN.