  The use of optimal transport cost for learning generative models has become
popular with Wasserstein Generative Adversarial Networks (WGAN). Training of
WGAN relies on a theoretical background: the calculation of the gradient of the
optimal transport cost with respect to the generative model parameters. We
first demonstrate that such gradient may not be defined, which can result in
numerical instabilities during gradient-based optimization. We address this
issue by stating a valid differentiation theorem in the case of entropic
regularized transport and specify conditions under which existence is ensured.
By exploiting the discrete nature of empirical data, we formulate the gradient
in a semi-discrete setting and propose an algorithm for the optimization of the
generative model parameters. Finally, we illustrate numerically the advantage
of the proposed framework.
