text: This very well written paper covers the span between W-GAN and VAE. For a reviewer who is not an expert in the domain, it reads very well, and would have been of tutorial quality if space had allowed for more detailed explanations. The appendix are very useful, and tutorial paper material (especially A). While I am not sure description would be enough to reproduce and no code is provided, every aspect of the architecture, if not described, if referred as similar to some previous work. There are also some notation shortcuts (not explained) in the proof of theorems that can lead to initial confusion, but they turn out to be non-ambiguous. One that could be improved is P(P_X, P_G) where one loses the fact that the second random variable is Y. This work contains plenty of novel material, which is clearly compared to previous work: - The main consequence of the use of Wasserstein distance is the surprisingly simple and useful Theorem 1. I could not verify its novelty, but this seems to be a great contribution. - Blending GAN and auto-encoders has been tried in the past, but the authors claim better theoretical foundations that lead to solutions that do not rquire min-max - The use of MMD in the context of GANs has also been tried. The authors claim that their use in the latent space makes it more practival The experiments are very convincing, both numerically and visually. Source of confusion: in algorithm 1 and 2, \tilde{z} is "sampled" from Q_TH(Z|xi), some one is lead to believe that this is the sampling process as in VAEs, while in reality Q_TH(Z|xi) is deterministic in the experiments.