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We provide non asymptotic rates of convergence of the Wasserstein GenerativeAdversarial networks (WGAN) estimator. We build neural networks classesrepresenting the generators and discriminators which yield a GAN that achievesthe minimax optimal rate for estimating a certain probability measure $\mu$with support in $\mathbb{R}^p$. The probability $\mu$ is considered to be thepush forward of the Lebesgue measure on the $d$-dimensional torus$\mathbb{T}^d$ by a map $g^\star:\mathbb{T}^d\rightarrow \mathbb{R}^p$ ofsmoothness $\beta+1$. Measuring the error with the $\gamma$-H\"older IntegralProbability Metric (IPM), we obtain up to logarithmic factors, the minimaxoptimal rate $O(n^{-\frac{\beta+\gamma}{2\beta +d}}\vee n^{-\frac{1}{2}})$where $n$ is the sample size, $\beta$ determines the smoothness of the targetmeasure $\mu$, $\gamma$ is the smoothness of the IPM ($\gamma=1$ is theWasserstein case) and $d\leq p$ is the intrinsic dimension of $\mu$. In theprocess, we derive a sharp interpolation inequality between H\"older IPMs. Thisnovel result of theory of functions spaces generalizes classical interpolationinequalities to the case where the measures involved have densities ondifferent manifolds.

Wasserstein GANs are Minimax Optimal Distribution Estimators