Why Adversarial Diffusion Trains More Stably Than GANs: A Local Jacobian View

Diffusion models are widely observed to train more reliably than GANs. We ask whether this stability comes primarily from the ELBO scalar objective or from the step-wise denoising structure itself. Building on the local dynamical-systems analysis of Mescheder et al. (2018), we show that min-max training induces rotational components, whereas the diffusion MSE objective yields a real spectrum near optima. For adversarial diffusion, the generator-discriminator coupling term is averaged across timesteps, reducing the effective rotational strength. Lightweight 2D experiments support the improved learning-rate robustness of adversarial diffusion over GANs, while ELBO-trained diffusion remains most stable in our setting.