Building on the success of deep learning, two modern approaches to learn a probability model of the observed data are Generative Adversarial Networks (GANs) and Variational AutoEncoders (VAEs). VAEs consider an explicit probability model for the data and compute a generative distribution by maximizing a variational lower-bound on the log-likelihood function. GANs, however, compute a generative model by minimizing a distance between observed and generated probability distributions without considering an explicit model for the observed data. The lack of having explicit probability models in GANs prohibits computation of sample likelihoods in their frameworks and limits their use in statistical inference problems. In this work, we show that an optimal transport GAN with the entropy regularization can be viewed as a generative model that maximizes a lower-bound on average sample likelihoods, an approach that VAEs are based on. In particular, our proof constructs an explicit probability model for GANs that can be used to compute likelihood statistics within GAN's framework. Our numerical results on several datasets demonstrate consistent trends with the proposed theory.
The contribution of the paper is to show that WGAN with entropic regularization maximize a lower bound on the likelihood of the observed data distribution. While the WGAN formulation minimizes the Wasserstein distance of the transformed latent distribution and the empirical distribution which is already a nice measure of "progress", having a bound on the likelihood can be interesting.
Pros:
+ I like the entropic GAN formulation and believe it is very interesting as it gives access to the joint distribution of latent and observed variables. 
+ While there are some doubtful statements, overall the paper is well written and easy to read.
Cons:
- The assumption of injectivity of the generator could be problematic, as it might not be fulfilled due to mode collapse.
- I feel the theory is not very deep. Since one has a closed form of the transportation map (Eq. 3.7), the likelihood of the data is obtained by marginalizing out the latent space. However, this assumes that the inner dual maximization problem is solved to stationarity so that Eq 3.7 holds, which is not the case in practice (5 discriminator updates).
- Thus in Sec. 4.1 for the likelihood at various points in training it is not clear what is actually happening.
- Sec 4.3 for unregularized GANs might be problematic. In general, the transportation plan is not a density function, so I'm not certain whether Theorem 1 / Corollary 2 still hold. Furthermore, the heuristic for "inverting" G^* is very crude. 
- There are also some minor problematic statements in the paper. While they can be easily fixed, they give me doubts:
  * The original VAE paper is not cited in the introduction for VAEs
  * The 2013 paper by Cuturi cited on page 2 has nothing to do with "computational aspects of GANs". It is about fast computation of approximate OT between two discrete prob. measures. 
  * First-order / second-order Wasserstein distance is I think a bit unusual name for W_1, W_2
  * On pg. 4, the point of the entropy term is to make the objective strongly convex. Strict convexity has no computational benefits.