Abstract: 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 joint training 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 that each generated point is influenced by at most one true point. The 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.