  Generative Adversarial Networks are a popular method for learning
distributions from data by modeling the target distribution as a function of a
known distribution. The function, often referred to as the generator, is
optimized to minimize a chosen distance measure between the generated and
target distributions. One commonly used measure for this purpose is the
Wasserstein distance. However, Wasserstein distance is hard to compute and
optimize, and in practice entropic regularization techniques are used to
improve numerical convergence. The influence of regularization on the learned
solution, however, remains not well-understood. In this paper, we study how
several popular entropic regularizations of Wasserstein distance impact the
solution in a simple benchmark setting where the generator is linear and the
target distribution is high-dimensional Gaussian. We show that entropy
regularization promotes the solution sparsification, while replacing the
Wasserstein distance with the Sinkhorn divergence recovers the unregularized
solution. Both regularization techniques remove the curse of dimensionality
suffered by Wasserstein distance. We show that the optimal generator can be
learned to accuracy $\epsilon$ with $O(1/\epsilon^2)$ samples from the target
distribution. We thus conclude that these regularization techniques can improve
the quality of the generator learned from empirical data for a large class of
distributions.
