  Generative modelling is often cast as minimizing a similarity measure between
a data distribution and a model distribution. Recently, a popular choice for
the similarity measure has been the Wasserstein metric, which can be expressed
in the Kantorovich duality formulation as the optimum difference of the
expected values of a potential function under the real data distribution and
the model hypothesis. In practice, the potential is approximated with a neural
network and is called the discriminator. Duality constraints on the function
class of the discriminator are enforced approximately, and the expectations are
estimated from samples. This gives at least three sources of errors: the
approximated discriminator and constraints, the estimation of the expectation
value, and the optimization required to find the optimal potential. In this
work, we study how well the methods, that are used in generative adversarial
networks to approximate the Wasserstein metric, perform. We consider, in
particular, the $c$-transform formulation, which eliminates the need to enforce
the constraints explicitly. We demonstrate that the $c$-transform allows for a
more accurate estimation of the true Wasserstein metric from samples, but
surprisingly, does not perform the best in the generative setting.
