Generative Adversarial Networks (GANs) have become a popular method to learn a probability model from data. Many GAN architectures with different optimization metrics have been introduced recently. Instead of proposing yet another architecture, this paper aims to provide an understanding of some of the basic issues surrounding GANs. First, we propose a natural way of specifying the loss function for GANs by drawing a connection with supervised learning. Second, we shed light on the generalization peformance of GANs through the analysis of a simple LQG setting: the generator is linear, the loss function is quadratic and the data is drawn from a Gaussian distribution. We show that in this setting: 1) the optimal GAN solution converges to population Principal Component Analysis (PCA) as the number of training samples increases; 2) the number of samples required scales exponentially with the dimension of the data; 3) the number of samples scales almost linearly if the discriminator is constrained to be quadratic. Thus, linear generators and quadratic discriminators provide a good balance for fast learning.
*Paper summary*
The paper considers GANs from a theoretical point of view. The authors approach GANs from the 2-Wasserstein point of view and provide several insights for a very specific setting. In my point of view, the main novel contribution of the paper is to notice the following fact:
(*) It is well known that the 2-Wasserstein distance W2(PY,QY) between multivariate Gaussian PY and its empirical version QY scales as $n^{-2/d}$, i.e. converges very slow as the dimensionality of the space $d$ increases. In other words, QY is not such a good way to estimate PY in this setting. A somewhat better way is use a Gaussian distribution PZ with covariance matrix S computed as a sample covariance of QY. In this case W2(PY, PZ) scales as $\sqrt{d/n}$.
The paper introduces this observation in a very strange way within the context of GANs. Moreover, I think the final conclusion of the paper (Eq. 19) has a mistake, which makes it hard to see why (*) has any relation to GANs at all.
There are several other results presented in the paper regarding relation between PCA and the 2-Wasserstein minimization for Gaussian distributions (Lemma 1 & Theorem 1). This is indeed an interesting point, however the proof is almost trivial and I am not sure if this provides any significant contribution for the future research.
Overall, I think the paper contains several novel ideas, but its structure requires a *significant* rework and in the current form it is not ready for being published. 
*Detailed comments*
In the first part of the paper (Section 2) the authors propose to use the optimal transport distance Wc(PY, g(PX)) between the data distribution PY (or its empirical version QY) and the model as the objective for GAN optimization. This idea is not novel: WGAN [1] proposed (and successfully implemented) to minimize the particular case of W1 distance by going through the dual form, [2] proposed to approach any Wc using auto-encoder reformulation of the primal (and also shoed that [5] is doing exactly W2 minimization), and [3] proposed the same using Sinkhorn algorithm. So this point does not seem to be novel.
The rest of the paper only considers 2-Wasserstein distance with Gaussian PY and Gaussian g(PX) (which I will abbreviate with R), which looks like an extremely limited scenario (and certainly has almost no connection to the applications of GANs).
Section 3 first establishes a relation between PCA and minimizing 2-Wasserstein distance for Gaussian distributions (Lemma 1, Theorem 1). Then the authors show that if R minimizes W2(PY, R) and QR minimizes W2(QY, QR) then the excess loss W2(PY, QR) - W2(PY, R) approaches zero at the rate $n^{-2/d}$ (both for linear and unconstrained generators). This result basically provides an upper bound showing that GANs need exponentially many samples to minimize W2 distance. I don't find these results novel, as they already appeared in [4] with a matching lower bound for the case of Gaussians (Theorem B.1 in Appendix can be modified easily to show this). As the authors note in the conclusion of Section 3, these results have little to do with GANs, as GANs are known to learn quite quickly (which contradicts the theory of Section 3).
Finally, in Section 4 the authors approach the same W2 problem from its dual form and notice that for the LQG model the optimal discriminator is quadratic. Based on this they reformulate the W2 minimization for LQG as the constrained optimization with respect to p.d. matrix A (Eq 16). The same conclusion does not work unfortunately for W2(QY, R), which is the real training objective of GANs. Theorem 3 shows that nevertheless, if we still constrain discriminator in the dual form of W2(QY, R) to be quadratic, the resulting soliton QR* performs the empirical PCA of Pn. 
This leads to the final conclusion of the paper, which I think contains a mistake. In Eq 19 the first equation, according to the definitions of the authors, reads
\[
W2(PY, QR) = W2(PY, PZ),   (**)
\]
where QR is trained to minimize min_R W2(QY, R) and PZ is as defined in (*) in the beginning of these notes. 
However, PZ is not the solution of min_R W2(QY, R) as the authors notice in the 2nd paragraph of page 8. Thus (**) is not true (at least, it is not proved in the current version of the text). PZ is a solution of min_R W2(QY, R) *where the discriminator is constrained to be quadratic*. This mismatch is especially strange, given the authors emphasize in the introduction that they provide bounds on divergences which are the same as used during the training (see 2nd paragraph on page 2) --- here the bound is on W2, but the empirical GAN actually does a regularized training (with constrained discriminator).
Finally, I don't think the experiments provide any convincing insights, because the authors use W1-minimization to illustrate properties of the W2. Essentially the authors say "we don't have a way to perform W2 minimization, so we rather do the W1 minimization and assume that these two are kind of similar".
* Other comments *
(1) Discussion in Section 2.1 seems to never play a role in the paper.
(2) Page 4: in p-Wasserstein distance, ||.|| does not need to be a Euclidean metric. It can be any metric.
(3) Lemma 2 seems to repeat the result from (Canas and Rosasco, 2012) as later cited by authors on page 7?
(4) It is not obvious how does Theorem 2 translate to the excess loss? 
(5) Section 4. I am wondering how exactly the authors are going to compute the conjugate of the discriminator, given the discriminator most likely is a deep neural network?
[1] Arjovsky et al., Wasserstein GAN, 2017
[2] Bousquet et al, From optimal transport to generative modeling: the VEGAN cookbook, 2017
[3] Genevay et al., Learning Generative Models with Sinkhorn Divergences, 2017
[4] Arora et al, Generalization and equilibrium in GANs, 2017
[5] Makhazani et al., Adversarial Autoencoders, 2015