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Generative Adversarial Networks (GANs) are a widely-used tool for generativemodeling of complex data. Despite their empirical success, the training of GANsis not fully understood due to the min-max optimization of the generator anddiscriminator. This paper analyzes these joint dynamics when the true samples,as well as the generated samples, are discrete, finite sets, and thediscriminator is kernel-based. A simple yet expressive framework for analyzingtraining called the $\textit{Isolated Points Model}$ is introduced. In theproposed model, the distance between true samples greatly exceeds the kernelwidth, so each generated point is influenced by at most one true point. Ourmodel enables precise characterization of the conditions for convergence, bothto good and bad minima. In particular, the analysis explains two common failuremodes: (i) an approximate mode collapse and (ii) divergence. Numericalsimulations are provided that predictably replicate these behaviors.

Instability and Local Minima in GAN Training with Kernel Discriminators