# A Convenient Infinite Dimensional Framework for Generative Adversarial Learning

arXiv: https://arxiv.org/abs/2011.12087 · HTML: https://mlsanity.com/paper/2011.12087/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2020.

## Built on by (most cited first, 11 of 11)

- [Generative Modeling of Turbulence](https://mlsanity.com/paper/2112.02548/) (2021)
- [Rates of convergence for density estimation with generative adversarial networks](https://mlsanity.com/paper/2102.00199/) (2021)
- [Detecting and Learning the Unknown in Semantic Segmentation](https://mlsanity.com/paper/2202.08700/) (2022)
- [Generalization capabilities of conditional GAN for turbulent flow under changes of geometry](https://mlsanity.com/paper/2302.09945/) (2023)
- [Numerical and statistical analysis of NeuralODE with Runge-Kutta time integration](https://mlsanity.com/paper/2503.10729/) (2025)
- [Learning Brenier Potentials with Convex Generative Adversarial Neural Networks](https://mlsanity.com/paper/2504.19779/) (2025)
- [When do World Models Successfully Learn Dynamical Systems?](https://mlsanity.com/paper/2507.04898/) (2025)
- [Comparison of Generative Learning Methods for Turbulence Surrogates](https://mlsanity.com/paper/2411.16417/) (2024)
- [Consistency of Learned Sparse Grid Quadrature Rules using NeuralODEs](https://mlsanity.com/paper/2507.01533/) (2025)
- [Regularity of Solutions to Beckmann's Parametric Optimal Transport](https://mlsanity.com/paper/2603.19755/) (2026)
- [Generative Adversarial Learning from Deterministic Processes](https://mlsanity.com/paper/2605.18425/) (2026)

## Abstract (arXiv)

In recent years, generative adversarial networks (GANs) have demonstrated impressive experimental results while there are only a few works that foster statistical learning theory for GANs. In this work, we propose an infinite dimensional theoretical framework for generative adversarial learning. We assume that the probability density functions of the underlying measure are uniformly bounded, k-times α-H"older differentiable (Ck,α) and uniformly bounded away from zero. Under these assumptions, we show that the Rosenblatt transformation induces an optimal generator, which is realizable in the hypothesis space of Ck,α-generators. With a consistent definition of the hypothesis space of discriminators, we further show that the Jensen-Shannon divergence between the distribution induced by the generator from the adversarial learning procedure and the data generating distribution converges to zero. Under certain regularity assumptions on the density of the data generating process, we also provide rates of convergence based on chaining and concentration.