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