Wasserstein GAN(WGAN) is a model that minimizes the Wasserstein distance between a data distribution and sample distribution. Recent studies have proposed stabilizing the training process for the WGAN and implementing the Lipschitz constraint. In this study, we prove the local stability of optimizing the simple gradient penalty $\mu$-WGAN(SGP $\mu$-WGAN) under suitable assumptions regarding the equilibrium and penalty measure $\mu$. The measure valued differentiation concept is employed to deal with the derivative of the penalty terms, which is helpful for handling abstract singular measures with lower dimensional support. Based on this analysis, we claim that penalizing the data manifold or sample manifold is the key to regularizing the original WGAN with a gradient penalty. Experimental results obtained with unintuitive penalty measures that satisfy our assumptions are also provided to support our theoretical results.
This paper deals with stability of simple gradient penalty $\mu$-WGAN optimization by introducing a concept of measure valued differentiation. WGAN with a squared zero centered gradient penalty term w.r.t. to a general measure is studied. Characterizes the convergence of gradient penalized Wasserstein GAN.