Paper:
Generative Adversarial Networks (GANs) are a very powerful framework for generative modeling. However, they are often hard to train, and learning of GANs often becomes unstable. Wasserstein GAN (WGAN) is a promising framework to deal with the instability problem as it has a good convergence property. One drawback of the WGAN is that it evaluates the Wasserstein distance in the dual domain, which requires some approximation, so that it may fail to optimize the true Wasserstein distance. In this paper, we propose evaluating the exact empirical optimal transport cost efficiently in the primal domain and performing gradient descent with respect to its derivative to train the generator network. Experiments on the MNIST dataset show that our method is significantly stable to converge, and achieves the lowest Wasserstein distance among the WGAN variants at the cost of some sharpness of generated images. Experiments on the 8-Gaussian toy dataset show that better gradients for the generator are obtained in our method. In addition, the proposed method enables more flexible generative modeling than WGAN.

Review:
The paper ‘Generative model based on minimizing exact empirical Wasserstein distance' proposes
a variant of Wasserstein GAN based on a primal version of the Wasserstein loss rather than the relying
on the classical Kantorovich-Rubinstein duality as first proposed by Arjovsky in the GAN context.
Comparisons with other variants of Wasserstein GAN is proposed on MNIST.
I see little novelty in the paper. The derivation of the primal version of the problem is already 
given in  
Cuturi, M., & Doucet, A. (2014, January). Fast computation of Wasserstein barycenters. In ICML (pp. 685-693).
Using optimal transport computed on batches rather the on the whole dataset is already used in (among
others)
 Genevay, A., Peyré, G., & Cuturi, M. (2017). Learning generative models with sinkhorn divergences. AISTATS
 Damodaran, B. B., Kellenberger, B., Flamary, R., Tuia, D., & Courty, N. (2018). DeepJDOT: Deep Joint distribution optimal transport for unsupervised domain adaptation. ECCV  
Also, the claim that the exact empirical Wasserstein distance is optimized is not true. The gradients, evaluated on 
batches, are biased. Unfortunately, the Wasserstein distance does not enjoy similar U-statistics as MMD. It is very 
well described in the paper (Section 3): 
https://openreview.net/pdf?id=S1m6h21Cb
Computing the gradients of Wasserstein on batches might be seen a kind of regularization, but it remains to be
proved and discussed.
Finally, the experimental validation appears insufficient to me (as only MNIST or toy datasets are considered).
Typos:
 Eq (1) and (2): when taken over the set of all Lipschitz-1 functions, the max should be a sup