# Nonparametric Density Estimation & Convergence Rates for GANs under Besov IPM Losses

arXiv: https://arxiv.org/abs/1902.03511 · HTML: https://mlsanity.com/paper/1902.03511/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2019.

## Built on by (most cited first, 22 of 22)

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- [Breaking the curse of dimensionality in structured density estimation](https://mlsanity.com/paper/2410.07685/) (2024)
- [Sobolev Space Regularised Pre Density Models](https://mlsanity.com/paper/2307.13763/) (2023)
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- [Generalized Balancing Weights via Deep Neural Networks](https://mlsanity.com/paper/2211.07533/) (2022)
- [Scalable Sobolev IPM for Probability Measures on a Graph](https://mlsanity.com/paper/2502.00737/) (2025)
- [Bona fide Riesz projections for density estimation](https://mlsanity.com/paper/2204.13606/) (2022)
- [Generalized Sobolev IPM for Graph-Based Measures](https://mlsanity.com/paper/2510.25591/) (2025)
- [Can Generative Artificial Intelligence Survive Data Contamination? Theoretical Guarantees under Contaminated Recursive Training](https://mlsanity.com/paper/2602.16065/) (2026)

## Abstract (arXiv)

We study the problem of estimating a nonparametric probability density under a large family of losses called Besov IPMs, which include, for example, Lp distances, total variation distance, and generalizations of both Wasserstein and Kolmogorov-Smirnov distances. For a wide variety of settings, we provide both lower and upper bounds, identifying precisely how the choice of loss function and assumptions on the data interact to determine the minimax optimal convergence rate. We also show that linear distribution estimates, such as the empirical distribution or kernel density estimator, often fail to converge at the optimal rate. Our bounds generalize, unify, or improve several recent and classical results. Moreover, IPMs can be used to formalize a statistical model of generative adversarial networks (GANs). Thus, we show how our results imply bounds on the statistical error of a GAN, showing, for example, that GANs can strictly outperform the best linear estimator.