Paper:
Stacked_WAE-2.pdf
Probabilistic models with hierarchical-latent-variable structures provide state-ofthe-art results amongst non-autoregressive, unsupervised density-based models. However, the most common approach to training such models based on Variational Autoencoders often fails to leverage deep-latent hierarchies; successful approaches require complex inference and optimisation schemes. Optimal Transport is an alternative, non-likelihood-based framework for training generative models with appealing theoretical properties, in principle allowing easier training convergence between distributions. In this work we propose a novel approach to training models with deep-latent hierarchies based on Optimal Transport, without the need for highly bespoke models and inference networks. We show that our method enables the generative model to fully leverage its deep-latent hierarchy, and that in-so-doing, it is more effective than the original Wasserstein Autoencoder with Maximum Mean Discrepancy divergence.

Review:
The paper aims to develop a deep generative model, which -unlike VAEs or GANs- comprises a hierarchy of latent variables rather than a direct map from the stochastic latent manifold to the observation space. To this end, the paper builds a training objective based on nesting the Wasserstein distance between the data distribution and its estimation arbitrarily many times. The generated objective corresponds naturally to a deep hierarchical generative model.
The principled approach followed to achieve the objective is solid and elegant. It is also intuitive and matches nicely with some valid observations highlighted in the paper such as insufficiency of by-passing intermediate latent variables (sentence above the Sec 2.3 title).
One major weakness of the paper is that it lacks a sufficient argumentation about how it differentiates from earlier attempts to nest Wasserstein distances. For instance,
Y. Dukler et al., "Wasserstein of Wasserstein Loss for Learning Generative Models", ICML, 2019
Apart from the theoretical argumentation, the paper should also compare their solution to this prior work on a number of benchmarks.
Another major weakness is that the paper lacks a quantitative evaluation scheme for its success. The experiments section starts with the claim that the proposed method "significantly" improves on the WAE, which I fail to see on the plots. 
Lastly, Having said that the proposed method is novel and elegant, it is still a straightforward extension of the existing and well-known Wasserstein Auto-Encoder (WAE) approach. It extends WAEs by repetitively applying the tricks proposed by this earlier work, putting aside some minor additional adjustments.
Minor on style: The abstract does not give any single hint about the methodological novelty of the work.
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Post-rebuttal: Thanks to authors for their effort for clarifications. Yet, I'm afraid the author response does not touch at all to any of the concerns I have raised. There are well-known ways to compare the success of generative models, FID being one of them as the authors point out. Another could be the test log-likelihood of a synthetic data set the true distribution of which can be predesigned. I understand the issues the authors raise about the difficulties in comparing generative models, but I kindly disagree with the attitude that there are no ways to compare, so we are obliged to live with qualitative comparisons. If a one-score comparison is not enough, the right way to go is to provide multiple scores. If direct metrics are not feasible, one should go for indirect ones, but should still provide outcomes a reader can reproduce.