Understanding the mathematical background of Generative Adversarial Networks (GANs)

Abstract

Generative Adversarial Networks (GANs) have gained widespread attention since their introduction, leading to numerous extensions and applications of the original GAN idea. A thorough understanding of GANs' mathematical foundations is necessary to use and build upon these techniques. However, most studies on GANs are presented from a computer science or engineering perspective, which can be challenging for beginners to understand fully. Therefore, this paper aims to provide an overview of the mathematical background of GANs, including detailed proofs of optimal solutions for vanilla GANs and boundaries for $f$-GANs that minimize a variational approximation of the $f$-divergence between two distributions. These contributions will enhance the understanding of GANs for those with a mathematical background and pave the way for future research.