# Dynamics of Fourier Modes in Torus Generative Adversarial Networks

arXiv: https://arxiv.org/abs/2209.01842 · HTML: https://mlsanity.com/paper/2209.01842/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 31 references that are in the map.
- First posted on arXiv 5 Sep 2022.

## Builds on (most cited first, 17 of 17)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [NIPS 2016 Tutorial: Generative Adversarial Networks](https://mlsanity.com/paper/1701.00160/) (2016)
- [Data Augmentation Generative Adversarial Networks](https://mlsanity.com/paper/1711.04340/) (2017)
- [Pros and Cons of GAN Evaluation Measures](https://mlsanity.com/paper/1802.03446/) (2018)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [Stabilizing Training of Generative Adversarial Networks through Regularization](https://mlsanity.com/paper/1705.09367/) (2017)
- [Gradient descent GAN optimization is locally stable](https://mlsanity.com/paper/1706.04156/) (2017)
- [GANS for Sequences of Discrete Elements with the Gumbel-softmax Distribution](https://mlsanity.com/paper/1611.04051/) (2016)
- [Evolutionary Generative Adversarial Networks](https://mlsanity.com/paper/1803.00657/) (2018)
- [Learning to Generate Images with Perceptual Similarity Metrics](https://mlsanity.com/paper/1511.06409/) (2015)
- [Importance Weighted Generative Networks](https://mlsanity.com/paper/1806.02512/) (2018)

## Abstract (arXiv)

Generative Adversarial Networks (GANs) are powerful Machine Learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable and typically it is necessary to implement several accessory heuristics to the networks to reach an acceptable convergence of the model. In this paper, we introduce a novel method to analyze the convergence and stability in the training of Generative Adversarial Networks. For this purpose, we propose to decompose the objective function of the adversary min-max game defining a periodic GAN into its Fourier series. By studying the dynamics of the truncated Fourier series for the continuous Alternating Gradient Descend algorithm, we are able to approximate the real flow and to identify the main features of the convergence of the GAN. This approach is confirmed empirically by studying the training flow in a 2-parametric GAN aiming to generate an unknown exponential distribution. As byproduct, we show that convergent orbits in GANs are small perturbations of periodic orbits so the Nash equillibria are spiral attractors. This theoretically justifies the slow and unstable training observed in GANs.