WASSERSTEIN GENERATIVE ADVERSARIAL NETWORKS ARE MINIMAX OPTIMAL DISTRIBUTION ESTIMATORS

• 成果类型: Article
• 署名作者: Stephanovitch, Arthur; Aamari, Eddie; Levrard, Clement
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS); Centre National de la Recherche Scientifique (CNRS); Universite de Rennes
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2430
• 发表日期: 2024
• 页码: 2167-2193
• 关键词: manifold

摘要

We provide nonasymptotic rates of convergence of the Wasserstein Generative Adversarial networks (WGAN) estimator. We build neural networks classes representing the generators and discriminators which yield a GAN that achieves the minimax optimal rate for estimating a certain probability measure mu with support in Rp. The probability mu is considered to be the push forward of the Lebesgue measure on the d-dimensional torus Td by a map g star : Td -> Rp of smoothness beta + 1. Measuring the error with the gamma-Holder Integral Probability Metric (IPM), we obtain up to logarithmic factors, the minimax optimal rate O (n termines the smoothness of the target measure mu , gamma is the smoothness of the IPM (gamma = 1 is the Wasserstein case) and d <= p is the intrinsic dimension of mu . In the process, we derive a sharp interpolation inequality between Holder IPMs. This novel result of theory of functions spaces generalizes classical interpolation inequalities to the case where the measures involved have densities on different manifolds.