We propose a general method for various image restoration problems, such as denoising, deblurring, super-resolution and inpainting. The problem is formulated as a constrained optimization problem. Its objective is to maximize a posteriori probability of latent variables, and its constraint is that the image generated by these latent variables must be the same as the degraded image. We use a Generative Adversarial Network (GAN) as our density estimation model. Convincing results are obtained on MNIST dataset.
The authors propose a method for image restoration, where the restored image is the MAP estimate. A pretrained GAN is utilized to approximate the prior distribution of the noise-free images. Then, the likelihood induces a constraint which is based on the degradation function. In particular, the method tries to find the latent point for which the GAN generates the image, which if gets degraded will match the given degraded image. Also, an optimization algorithm is presented that solves the proposed constrained optimization problem.
I find the paper very well written and easy to follow. Also, the idea is pretty clean, and the derivations are simple and clear. Additionally, the Figures 2,3 are very intuitive and nicely explain the theory. However, I think that there are some weaknesses (see comments):
Comments: 
#1) I do not understand exactly what the "general method" means. Does it mean that you propose a method, where you can just change the F, such that to solve a different degradation problem? So you provide the general framework where somebody has to specify only the F?
#2) Clearly, the efficiency of the method is highly based on the ability of the GAN to approximate well the prior distribution of the noise-free images.
#3) There are several Equations that can be combined, such that to save enough white space in order to discuss further some actual technical details. For instance, Eq. 2,3 can be easily combined using the proportional symbol, Eq. 8,9,10,11 show actually the same thing.
#4) I think that the function F has to be differentiable, and this should be mentioned in the text. Also, I believe that some actual (analytic) examples of F should be provided, at least  in the experiments. The same holds for the p(Omega). This parameter Omega is estimated individually for each degraded image?
#5) Before Eq. 8 the matrix V is a function of z and should be presented as such in the equations.
#6) I believe that it would be nice to include a magnified image of Fig. 3, where the gradient steps are shown. Also, my understanding is that the optimization goal is to find first a feasible solution, and then find the point that maximizes f. I think that this can be clarified in the text.
#7) The optimization steps seem to be intuitive, however, there is not any actual proof of converge. Of course, the example in the Figure 3 is very nice and intuitive, but it is also rather simple. I would suggest, at least, to include some empirical evidences in the experiments that show convergence.
#8) In the experiments I think that at least one example of F and p(Omega) should be presented. Also, what the numbers in Table 4 show? Which is the best value that can be achieved? These numbers correspond to several images, or to a unique image? 
#9) I think that MNIST is almost a toy experiment, since the crucial component of the proposed method is the prior modeling with the GAN. I believe that a more challenging experiment should be conducted e.g. using celebA dataset.
Minor comments:
#1) In the paragraph after Eq. 4 the equality p_r(x)=p_G(x) is very strong assumption. I would suggest to use the \simeq symbol instead.
#2) After Eq. 6 the "nonnegative" should be "nonzero".
#3) Additional density estimation models can be used e.g. VAEs, GMM. Especially, I believe that the VAE will provide a way to approximate the prior easier than the GAN.
#4) In Section 2 paragraph 2, the sentence "However, they only ... and directly" is not clear what means.
In general, I find both the proposed model and optimization algorithm interesting. Additionally, the idea is nicely presented in the paper. Most of my comments are improvements which can be easily included. The two things that make me more skeptical, is the convergence of the proposed algorithm and the experiments. The MNIST is a relatively simple experiment, and I would like to see how the method works in more challenging problems. Also, I think that additional methods to compute the image prior should be included in the experiments.