Paper:
Published as a conference paper at ICLR 2018 LEARNING WASSERSTEIN EMBEDDINGS
The Wasserstein distance received a lot of attention recently in the community of machine learning, especially for its principled way of comparing distributions. It has found numerous applications in several hard problems, such as domain adaptation, dimensionality reduction or generative models. However, its use is still limited by a heavy computational cost. Our goal is to alleviate this problem by providing an approximation mechanism that allows to break its inherent complexity. It relies on the search of an embedding where the Euclidean distance mimics the Wasserstein distance. We show that such an embedding can be found with a siamese architecture associated with a decoder network that allows to move from the embedding space back to the original input space. Once this embedding has been found, computing optimization problems in the Wasserstein space (e.g. barycenters, principal directions or even archetypes) can be conducted extremely fast. Numerical experiments supporting this idea are conducted on image datasets, and show the wide potential benefits of our method.

Review:
This paper proposes approximating the Wasserstein distance between normalized greyscale images based on a learnable approximately isometric embedding of images into Euclidean space. The paper is well written with clear and generally thorough prose. It presents a novel, straightforward and practical solution to efficiently computing Wasserstein distances and performing related image manipulations.
Major comments:
It sounds like the same image may be present in the training set and eval set. This is methodologically suspect, since the embedding may well work better for images seen during training. This affects all experimental results.
I was pleased to see a comparison between using exact and approximate Wasserstein distances for image manipulation in Figure 5, since that's a crucial aspect of whether the method is useful in practice. However the exact computation (OT LP) appears to be quite poor. Please explain why the approximation is better than the exact Wasserstein difference for interpolation. Relatedly, please summarize the argument in Cuturi and Peyre that is cited ("as already explained in").
Minor comments:
In section 3.1 and 4.1, "histogram" is used to mean normalized-to-sum-to-1 images, which is not the conventional meaning.
It would help to pick one of "Wasserstein Deep Learning" and "Deep Wasserstein Embedding" and use it and the acronym consistently throughout.
"Disposing of a decoder network" in section 3.1 should be "using a decoder network"?
In section 4.1, the architectural details could be clarified. What size are the input images? What type of padding for the convolutions? Was there any reason behind the chosen architecture? In particular the use of a dense layers followed by convolutional layers seems peculiar.
It would be helpful to say explicitly what "quadratic ground metric" means (i.e. W_2, I presume) in section 4.2 and elsewhere.
It would be helpful to give a sense of scale for the numbers in Table 1, e.g. give the 95th percentile Wasserstein distance. Perhaps use the L2 distance passed through a 1D-to-1D learned warping as a baseline.
Mention that OT stands for optimal transport in section 4.3.
Suggest mentioning "there is no reason for a Wasserstein barycenter to be a realistic sample" in the main text when first discussing barycenters.