Assessing distance between the true and the sample distribution is a key component of many state of the art generative models, such as Wasserstein Autoencoder (WAE). Inspired by prior work on Sliced-Wasserstein Autoencoders (SWAE) and WAE-MMD (WAE using maximum mean discrepancy based distance function) we construct a new generative model -Cramer-Wold AutoEncoder (CWAE). CWAE cost function is based upon a characteristic kernel we introduce, called CramerWold kernel, and has a a simple closed-form in the case of normal prior. As a consequence, while simplifying the optimization procedure (no need of sampling necessary to evaluate the distance function in the training loop), CWAE performance matches quantitatively and qualitatively that of WAE-MMD and often improves upon SWAE.
The paper introduces a novel regularized auto-encoder architecture called the Cramer-Wold AutoEncoders (CWAE). It's objective (Eq. 7) consists of two terms: (i) a standard reconstruction term making sure the the encoder-decoder pair aligns nicely to accurately reconstruct all the training images and (ii) the regularizer, which roughly speaking requires the encoded training distribution to look similar to the standard normal (which is a prior used in the generative model being trained). The main novelty of the paper is in the form of this regularizer. The authors introduce what they call "the Cramer-Wold distance" (for definitions see Theorems 3.1 and 3.2) which is defined between two finite sets of D-dimensional points. The authors provide empirical studies showing that the proposed CWAE method achieves the same quality of samples (measured with FID scores) as the WAE-MMD model [1] previously reported in the literature, while running faster (by up to factor of 2 reduction in the training time, as the authors report). 
While on the AE model / architecture side I feel the contribution is very marginal, I still think that the improvement in the training speed is something useful. Otherwise it is a nicely written and polished piece of work. 
Detailed comments:
(1) My main problem with this paper is that the novel objective proposed by the authors in Eq. 7 is equivalent to the objective of WAEs appearing in Eq. 4 of [1] (up to a heuristic of applying logarithm to the divergence measure, which is not justified but meant to "improve the balance between two terms", see footnote 2), where the authors use the newly introduced Cramer-Wold divergence as a choice of the penalty term in Eq. 4 of [1]. 
(2) When viewed in this way, CW-distance introduced in Eq (2) closely resembles the unbiased U-statistic estimate of the MMD used in WAE-MMD [1, Algorithm 2]. In other words: it may be the case that there is a choice of a reproducing kernel k such that Eq. 2 of this paper is an estimate of MMD_k between two distributions based on the i.i.d. samples X and Y. Note that if it is indeed the case, this corresponds to the V-statistic and thus biased: in U-statistic the diagonal terms (that is i = i' and j = j' in forst two terms of eq 2) would be omitted. If this all is indeed the case, it is not surprising that the numbers the authors get in the experiments are so similar to WAE-MMD, because CWAE would be exactly WAE-MMD with a specific choice of the kernel. 
(3) The authors make a big deal out of their proposed divergence measure not requiring samples from the prior as opposed to WAE-MMD. However, WAE-MMD does not necessarily need to sample from the prior when used with Gaussian prior and Gaussian RBF kernel, because in this case the prior-related parts of the MMD can be computed analytically.  In other words, if the computational advantage of CWAE compared to WAE-MMD comes from CWAE not sampling Pz, the computational overhead of WAE-MMD can be eliminated at least in the above-mentioned setting.
(4) based on the name "CW distance" I would expect the authors to actually prove that it is indeed a distance (i.e. all the main axioms). 
(5) The authors override the CW distance: first in Theorem 3.1 they define it as a distance between two finite point clouds, and later in Theorem 3.2 they redefine it as a distance between a point cloud and the Gaussian distribution. 
(6) What is image(X) in Remark 4.1?
[1] Tolstikhin et al., Wasserstein Auto-Encoders, 2017.