Overall, the contributions of this paper are clear. The theoretical derivations seem correct. Also, some experiments validate the related claims in the paper.

This paper builds on ideas from Optimal transport to derive an objective for the discriminator that the authors claim is an improvement over the Wasserstein GAN. A formulation for the GAN discriminator objective is set up using Optimal Transport for 1-Wasserstein distance. This is obtained by examining transport in the Monge Problem and a proof is provided for a minimizer in this context. The Lipshitz constant K is estimated from data. The authors claim that it does not vary much, and therefore can be assumed constant. Recipes are provided for calculations in data space (Algorithm 1) and latent space (Algorithm 2). The results are quite reasonable, with examples given for images (CIFAR-10, SLT-10) and the swiss roll dataset. The paper claims that improvements are obtained over WGAN-GP (after training it for some epochs), SN-GAN. The authors' rebuttal has mostly convinced me that the formulation is sound. Most of my initial doubts had to do with the assumption that the automorphism (equation (7) in the appendix) exists, which is necessary to complete the proof that sets up the objective function. The authors draw attention to works [1] and [2] for insight. [1] L. Caffarelli, M. Feldman, and R. McCann. Journal of the American Mathematical Society, 15(1):1–26, 2002. [2] W. Gangbo and R. McCann. Acta Mathematica, 177(2):113–161, 1996. Improvements: 1) I am hoping that the language issues can be improved. 2) An explanation detailing the correctness of the assumptions pertaining to the existence of the solution to the Monge Problem (theorem 1) would be most helpful in making the paper readable. Overall: I rule this as an accept. Good paper, but the clarity could be improved. Quality: Good Clarity: Unclear. Significance: The work is highly significant in that it is of interest that we widen our theoretical and application apparatus for insight and to improve the quality of existing GAN results.

The paper makes an original contribution to the field of generative modelling that is backed by a strong theoretical analysis. It is generally written in a readable way, even though some parts could be smoothed out a little. The experiments conducted are sufficient to prove the merits of the proposed method. I think the contributions will have some significance in the field of generative modelling, even though some practical problems (i.e. runtime) are not discussed in much detail.