  The mathematical forces at work behind Generative Adversarial Networks raise
challenging theoretical issues. Motivated by the important question of
characterizing the geometrical properties of the generated distributions, we
provide a thorough analysis of Wasserstein GANs (WGANs) in both the finite
sample and asymptotic regimes. We study the specific case where the latent
space is univariate and derive results valid regardless of the dimension of the
output space. We show in particular that for a fixed sample size, the optimal
WGANs are closely linked with connected paths minimizing the sum of the squared
Euclidean distances between the sample points. We also highlight the fact that
WGANs are able to approach (for the 1-Wasserstein distance) the target
distribution as the sample size tends to infinity, at a given convergence rate
and provided the family of generative Lipschitz functions grows appropriately.
We derive in passing new results on optimal transport theory in the
semi-discrete setting.
