# Hellinger loss function for Generative Adversarial Networks

arXiv: https://arxiv.org/abs/2512.12267 · HTML: https://mlsanity.com/paper/2512.12267/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 18 references that are in the map.
- First posted on arXiv 13 Dec 2025.

## Builds on (most cited first, 11 of 11)

- [Least Squares Generative Adversarial Networks](https://mlsanity.com/paper/1611.04076/) (2016)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Some Theoretical Properties of GANs](https://mlsanity.com/paper/1803.07819/) (2018)
- [Ten Years of Generative Adversarial Nets (GANs): A survey of the state-of-the-art](https://mlsanity.com/paper/2308.16316/) (2023)
- [On the Statistical Properties of Generative Adversarial Models for Low Intrinsic Data Dimension](https://mlsanity.com/paper/2401.15801/) (2024)
- [Cumulant GAN](https://mlsanity.com/paper/2006.06625/) (2020)
- [Generative Adversarial Nets for Robust Scatter Estimation: A Proper Scoring Rule Perspective](https://mlsanity.com/paper/1903.01944/) (2019)
- [Robust Generative Adversarial Network](https://mlsanity.com/paper/2004.13344/) (2020)
- [Robust Estimation for Nonparametric Families via Generative Adversarial Networks](https://mlsanity.com/paper/2202.01269/) (2022)
- [zGAN: An Outlier-focused Generative Adversarial Network For Realistic Synthetic Data Generation](https://mlsanity.com/paper/2410.20808/) (2024)

## Abstract (arXiv)

We propose Hellinger-type loss functions for training Generative Adversarial Networks (GANs), motivated by the boundedness, symmetry, and robustness properties of the Hellinger distance. We define an adversarial objective based on this divergence and study its statistical properties within a general parametric framework. We establish the existence, uniqueness, consistency, and joint asymptotic normality of the estimators obtained from the adversarial training procedure. In particular, we analyze the joint estimation of both generator and discriminator parameters, offering a comprehensive asymptotic characterization of the resulting estimators. We introduce two implementations of the Hellinger-type loss and we evaluate their empirical behavior in comparison with the classic (Maximum Likelihood-type) GAN loss. Through a controlled simulation study, we demonstrate that both proposed losses yield improved estimation accuracy and robustness under increasing levels of data contamination.