Rates of convergence for nonparametric estimation of singular distributions using generative adversarial networks

We consider generative adversarial networks (GAN) for estimating parameters in a deep generative model. The data-generating distribution is assumed to concentrate around some low-dimensional structure, making the target distribution singular to the Lebesgue measure. Under this assumption, we obtain convergence rates of a GAN type estimator with respect to the Wasserstein metric. The convergence rate depends only on the noise level, intrinsic dimension and smoothness of the underlying structure. Furthermore, the rate is faster than that obtained by likelihood approaches, which provides insights into why GAN approaches perform better in many real problems. A lower bound of the minimax optimal rate is also investigated.