Mixture Model (MM) is a probabilistic framework which allows us to define a dataset containing K different modes.
This paper proposes a modification of GANs where the latent space follows a distribution modelled by a Gaussian Mixture Model. While the idea of using GMMs in GANs is not novel, the main contribution of the paper is to add a classification models that enables posterior inference. The whole model is trained jointly end-to-end using both an adversarial loss and a mutual information loss. The procedure is then tested on MNIST, Fashion-MNIST and a subset of Oxford-102 Flower. 
While the problem of posterior inference is interesting, the novelty of the paper is quite limited. The overall structure is clear, although writing can be improved. 
However, a part from the form, I have other concerns about the paper: 
- The main purpose of the paper is to tackle large image datasets that are hard to address by classical GMMs. This being said, the used datasets are composed of small and modal enough images that it seems hard to validate the claim of the paper using only these data. It seems to me that for the claim of the paper to be verified, larger scale/more complex datasets are needed. 
- In all experiments, the authors suppose they have access to the number of classes/modes in the data, which is a huge assumption. It would be interesting to see if it would be possible to automatically accurately select the number of modes, e.g. on a held out validation set.
- One problem of GANs that the authors do not seem to consider is mode collapse. It would be interesting to do experiments with unbalanced datasets (e.g. MNIST 1 vs all) to see if the proposed architecture will model the data correctly. 
- I am confused about the use of Mutual Information loss. The author claim that they would like to enforce each of the generated images to be from the same class as the input image. This would make sense if the authors used the multinomial sampling for the latent variable generation. However, the K generated samples are from K different Gaussians. It seems unreasonable to require of the classifier to render the same result. 
- On the same note, in order to both use the classical multinomial sampling in GMMs and not break the backpropagation, have the authors considered updating the classifier and the GAN in separately in expectation-maximization fashion?
- Finally, I don't see the point of weighting the adversarial loss by the weights of Gaussians. All the generated images are fake and should be equally detected as such. 
Although the idea is interesting, I think the paper, at its current status, not ready for publication. 
Minor: 
- P. 4: Generator: The sampling density from the multinomial distribution seems incorrect. However, as the authors skip the sampling step to be able to back propagate through the model, this is not significant. 
- P. 1: The paragraph before the last: may even synthesizing -> synthesize
- P. 3: Architecture: Possibility -> Probability
- P. 7: Figure title: CIFR10 -> Oxford-102
- Algorithm notation: 
    *Indexes for alpha and alpha_hat from 1 and not 0 to be consistent with the rest of the text 
    *Add hats to the entries of alpha_hat
    *  LI from alpha_i and alpha_hat_i as in Equation 4 -> maybe change alpha_hat by x_hat to be consistent with the notation of eq.4 (although the meaning is clear here).