Current generative models like generative adversarial networks (GANs) and variational autoencoders (VAEs) have attracted huge attention due to its capability to generate visual realistic images. However, most of the existing models suffer from the mode collapse or mode mixture problems. In this work, we give a theoretic explanation of the both problems by Figalli's regularity theory of optimal transportation maps. Basically, the generator compute the transportation maps between the white noise distributions and the data distributions, which are in general discontinuous. However, deep neural networks (DNNs) can only represent continuous maps. This intrinsic conflict induces mode collapse and mode mixture. In order to tackle the both problems, we explicitly separate the manifold embedding and the optimal transportation; the first part is carried out using an autoencoder (AE) to map the images onto the latent space; the second part is accomplished using a GPU-based convex optimization to find the discontinuous transportation maps. Composing the extended optimal transport (OT) map and the decoder, we can finally generate new images from the white noise. This AE-OT model avoids representing discontinuous maps by DNNs, therefore effectively prevents mode collapse and mode mixture.
General Comments:  The generator in Generative Adversarial Networks (GANS) computes an optimal transportation from the noise distribution to the data distribution.  However, such maps are in general discontinuous.  Since deep neural networks can only represent continuous maps, this brings two problems: mode collapse and mode mixture. This paper approaches both problems using Figalli's regularity theory. They separate the manifold embedding (here an autoencoder maps input data to a latent space) from the optimal transportation (this map is found by convex optimization). Composing these two steps yields the proposed method. Their method basically avoids representing discontinuous maps by the generator. Empirically, the proposed method performs similar or better than state-of-the-art.
I think the idea of the paper is nice, and an interesting perspective  on GANs is presented. A new method is proposed. The numerical contributions are certainly significant. Therefore, I believe the paper deserves publication.
Nevertheless, I have some comments below.
1) Although this paper brings a new perspective, based on optimal transport theory, as far as I can understand this paper does not establish formal new results. Thus I think some strong claims about providing deep theoretical explanation should be more moderate. In essence, it seems that the paper verifies *numerically* (in section B.3) that Figalli's theorem (stated in Appendix B) holds in this context.
2) This is just a suggestion. I think in some parts a lighter notation and a more intuitive explanation could help.
3) After Eq. (5) in the Appendix the authors mention Newton's method, and Thm 3 is also specific to Newton's method. Then they mention that *Gradient Descent* is used (and in the main part of the paper they mentioned Adam). This is confusing. All these algorithms are different, and Newton's method does not imply convergence results for gradient descent. I don't see how Thm 3 is relevant.
4) This is a simple doubt. To avoid non-differentiability of the gradient, the OT step computes the Brenier potential and is able to locate the singularities. I wonder if using a simpler approach through optimization for nosmooth problems (such as Moreau envelopes or proximal methods) could resolve this issue? In the negative case, why not?
5) Some Minor comments:
1. Define OT in the abstract (Optimal Transportation?) 
2. What is AE? (not defined also; Auto Encoder?)
3. There are lots of typos through the text, such as missing "the", "a", etc. 
and a couple mispelled words. I suggest the authors proofread the draft
more carefully.
4. pp. 4 ... what is a "PL convex function". PL is not defined.