Understanding and improving Generative Adversarial Networks (GAN) using notions from Optimal Transport (OT) theory has been a successful area of study, originally established by the introduction of the Wasserstein GAN (WGAN). An increasing number of GANs incorporate OT for improving their discriminators, but that is so far the sole way for the two domains to cross-fertilize. In this work we address the converse question: is it possible to recover an optimal map in a GAN fashion? To achieve this, we build a new model relying on the second Wasserstein distance. This choice enables the use of many results from OT community. In particular, we may completely describe the dynamics of the generator during training. In addition, experiments show that practical uses of our model abide by the rule of evolution we describe. As an application, our generator may be considered as a new way of computing an optimal transport map. It is competitive in low-dimension with standard and deterministic ways to approach the same problem. In high dimension, the fact it is a GAN-style method makes it more powerful than other methods.
The paper W2GAN describes a method for training GAN and computing an optimal transport (OT) map
between distributions. As far as I can tell, it is difficult to identify the original contributions
of the paper. Most results are known from the OT community. The differences with the work of Seguy, 2018
is also not obvious. I encourage the authors to establish more clearly the differences of their work
with this last reference. Most of the theoretical considerations of Section 3 is either based on 
unrealistic assumptions (case 1) or make vague assumptions 'if we ignore the possibly significant effect ...'
that seem unjustified so far. Experimental results do not show evidences of superiority wrt. existing works.  
All in all I would recommend the authors to better focus on the original contribution of their works wrt.  
state-of-the-art and explain why the theoretical analysis on convergence following a geodesic path in a 
Wasserstein space is valuable from a practical view. Finally, I did not understand the final claim of the 
Abstract : 'Perhaps surprisingly, we also provide empirical evidence that other GANs also approximately following
the Optimal Transport.'. What are those empirical evidences ? It seems that this claim is not supported somewhere 
else in the paper.
Minor remarks:
 - regarding the penalization in eq. (5), the expectation is not for all x and y \in R^2, but for x drawn from \mu and y from \nu.
   Same for L_2 regularization
 - Proposition 1 is mainly due to Brenier
Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector‐valued functions. Communications on pure and applied mathematics, 44(4), 375-417.
 - from Eq (7), you should give precisely over what the expectations are taken.
 - Eq (10) : how do you inverse sup and inf ? 
 - when comparing to Seguy 2018, are you using an entropic or a L_2 regularization ? How do you set the regularization strength ?
 - where is Figure 2.a described in section 4.2 ? 
Related works :
 - what is reference (Alexandre, 2018) ?  
 - regarding applications of OT to domain adaptation, there are several references on the subject. 
   See for instance 
Courty, N., Flamary, R., Tuia, D., & Rakotomamonjy, A. (2017). Optimal transport for domain adaptation. IEEE transactions on pattern analysis and machine intelligence, 39(9), 1853-1865.
or 
Damodaran, B. B., Kellenberger, B., Flamary, R., Tuia, D., & Courty, N. (2018). DeepJDOT: Deep Joint distribution optimal transport for unsupervised domain adaptation. ECCV 
for a deep variant.
 - Reference Seguy 2017 and 2018 are the same and should be fused. The corresponding paper
   was published at ICLR 2018
   Regarding this last reference, the claim 'As far as we know, it is the first demonstration of a GAN achieving reasonable generative modeling results and an approximation of the optimal transport map between two continuous distributions.' should maybe be lowered ?