# Sobolev GAN

arXiv: https://arxiv.org/abs/1711.04894 · HTML: https://mlsanity.com/paper/1711.04894/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Nov 2017.

## Built on by (most cited first, 29 of 29)

- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [An error analysis of generative adversarial networks for learning distributions](https://mlsanity.com/paper/2105.13010/) (2021)
- [Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives](https://mlsanity.com/paper/2305.08466/) (2023)
- [Approximation bounds for norm constrained neural networks with applications to regression and GANs](https://mlsanity.com/paper/2201.09418/) (2022)
- [VoLTA: Vision-Language Transformer with Weakly-Supervised Local-Feature Alignment](https://mlsanity.com/paper/2210.04135/) (2022)
- [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022)
- [Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss](https://mlsanity.com/paper/2402.00152/) (2024)
- [MonoFlow: Rethinking Divergence GANs via the Perspective of Wasserstein Gradient Flows](https://mlsanity.com/paper/2302.01075/) (2023)
- [An Optimal Transport Perspective on Unpaired Image Super-Resolution](https://mlsanity.com/paper/2202.01116/) (2022)
- [Sample Complexity Bounds for Estimating Probability Divergences under Invariances](https://mlsanity.com/paper/2311.02868/) (2023)
- [Realizing GANs via a Tunable Loss Function](https://mlsanity.com/paper/2106.05232/) (2021)
- [Multi-Discriminator Sobolev Defense-GAN Against Adversarial Attacks for End-to-End Speech Systems](https://mlsanity.com/paper/2103.08086/) (2021)
- [A Likelihood Based Approach to Distribution Regression Using Conditional Deep Generative Models](https://mlsanity.com/paper/2410.02025/) (2024)
- [Towards Robust Speech-to-Text Adversarial Attack](https://mlsanity.com/paper/2103.08095/) (2021)
- [Bi-Discriminator Class-Conditional Tabular GAN](https://mlsanity.com/paper/2111.06549/) (2021)
- [Adaptive Consensus Optimization Method for GANs](https://mlsanity.com/paper/2304.10317/) (2023)
- [Hoeffding's Inequality for Markov Chains under Generalized Concentrability Condition](https://mlsanity.com/paper/2310.02941/) (2023)
- [Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances](https://mlsanity.com/paper/2310.13433/) (2023)
- [Integral Probability Metrics Meet Neural Networks: The Radon-Kolmogorov-Smirnov Test](https://mlsanity.com/paper/2309.02422/) (2023)
- [Elastic Interaction Energy-Based Generative Model: Approximation in Feature Space](https://mlsanity.com/paper/2303.10553/) (2023)
- [Generative adversarial learning with optimal input dimension and its adaptive generator architecture](https://mlsanity.com/paper/2405.03723/) (2024)
- [Scalable Sobolev IPM for Probability Measures on a Graph](https://mlsanity.com/paper/2502.00737/) (2025)
- [ReLU integral probability metric and its applications](https://mlsanity.com/paper/2504.18897/) (2025)
- [An Empirical Study on GANs with Margin Cosine Loss and Relativistic Discriminator](https://mlsanity.com/paper/2110.11293/) (2021)
- [RCC-GAN: Regularized Compound Conditional GAN for Large-Scale Tabular Data Synthesis](https://mlsanity.com/paper/2205.11693/) (2022)
- [RSD-GAN: Regularized Sobolev Defense GAN Against Speech-to-Text Adversarial Attacks](https://mlsanity.com/paper/2207.06858/) (2022)
- [The informativeness of the gradient revisited](https://mlsanity.com/paper/2505.22158/) (2025)
- [Generalized Sobolev IPM for Graph-Based Measures](https://mlsanity.com/paper/2510.25591/) (2025)

## Abstract (arXiv)

We propose a new Integral Probability Metric (IPM) between distributions: the Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions for functions (critic) restricted to a Sobolev ball defined with respect to a dominant measure μ. We show that the Sobolev IPM compares two distributions in high dimensions based on weighted conditional Cumulative Distribution Functions (CDF) of each coordinate on a leave one out basis. The Dominant measure μ plays a crucial role as it defines the support on which conditional CDFs are compared. Sobolev IPM can be seen as an extension of the one dimensional Von-Mises Cram'er statistics to high dimensional distributions. We show how Sobolev IPM can be used to train Generative Adversarial Networks (GANs). We then exploit the intrinsic conditioning implied by Sobolev IPM in text generation. Finally we show that a variant of Sobolev GAN achieves competitive results in semi-supervised learning on CIFAR-10, thanks to the smoothness enforced on the critic by Sobolev GAN which relates to Laplacian regularization.