Paper:
Reliable measures of statistical dependence could be useful tools for learning independent features and performing tasks like source separation using Independent Component Analysis (ICA). Unfortunately, many of such measures, like the mutual information, are hard to estimate and optimize directly. We propose to learn independent features with adversarial objectives (Goodfellow et al., 2014;Huszar, 2016) which optimize such measures implicitly. These objectives compare samples from the joint distribution and the product of the marginals without the need to compute any probability densities. We also propose two methods for obtaining samples from the product of the marginals using either a simple resampling trick or a separate parametric distribution. Our experiments show that this strategy can easily be applied to different types of model architectures and solve both linear and non-linear ICA problems.

Review:
The focus of the paper is independent component analysis (ICA) and its nonlinear variants such as the post non-linear (PNL) ICA model. Motivated by the fact that estimating mutual information and similar dependency measures require density estimates and hard to optimize, the authors propose a Wasserstein GAN (generative adversarial network) based solution to tackle the problem, with illustrations on 6 (synthetic) and 3-dimemensional (audio) examples. The primary idea of the paper is to use the Wasserstein distance as an independence measure of the estimated source coordinates, and optimize it in a neural network (NN) framework.
Although finding novel GAN applications is an exciting topic, I am not really convinced that ICA with the proposed Wasserstein GAN based technique fulfills this goal.
 
Below I detail my reasons:
1)The ICA problem can be formulated as the minimization of pairwise mutual information [1] or one-dimensional entropy [2]. In other words, estimating the joint dependence of the source coordinates is not necessary; it is worthwhile to avoid it.
2)The PNL ICA task can be efficiently tackled by first 'removing' the nonlinearity followed by classical linear ICA; see for example [3].
3)Estimating information theoretic (IT) measures (mutual information, divergence) is a quite mature field with off-the-self techniques, see for example [4,5,6,8]. These methods do not estimate the underlying densities; it would be superfluous (and hard).
4)Optimizing non-differentiable IT measures can computationally quite efficiently carried out in the ICA context by e.g., Givens rotations [7]; differentiable ICA cost functions can be robustly handled by Stiefel manifold methods; see for example [8,9].
5)Section 3.1: This section is devoted to generating samples from the product of the marginals, even using separate generator networks. I do not see the necessity of these solutions; the subtask can be solved by independently shuffling all the coordinates of the sample.
6)Experiments (Section 6): 
i) It seems to me that the proposed NN-based technique has some quite serious divergence issues: 'After discarding diverged models, ...' or 'Unfortunately, the model selection procedure also didn't identify good settings for the Anica-g model...'.
ii) The proposed method gives pretty comparable results to the chosen baselines (fastICA, PNLMISEP) on the selected small-dimensional tasks. In fact, [7,8,9] are likely to provide more accurate (fastICA is a simple kurtosis based method, which is 
a somewhat crude 'estimate' of entropy) and faster estimates; see also 2).
References:
[1] Pierre Comon. Independent component analysis, a new concept? Signal Processing, 36:287-314, 1994.
[2] Aapo Hyvarinen and Erkki Oja. Independent Component Analysis: Algorithms and Applications. Neural Networks, 13(4-5):411-30, 2000. 
[3] Andreas Ziehe, Motoaki Kawanabe, Stefan Harmeling, and Klaus-Robert Muller. Blind separation of postnonlinear mixtures using linearizing transformations and temporal decorrelation. Journal of Machine Learning Research, 4:1319-1338, 2003.
[4] Barnabas Poczos, Liang Xiong, and Jeff Schneider. Nonparametric divergence: Estimation with applications to machine learning on distributions. In Conference on Uncertainty in Artificial Intelligence, pages 599-608, 2011.
[5] Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Scholkopf, Alexander Smola. A Kernel Two-Sample Test. Journal of Machine Learning Research, 13:723-773, 2012.
[6] Alan Wisler, Visar Berisha, Andreas Spanias, Alfred O. Hero. A data-driven basis for direct estimation of functionals of distributions. TR, 2017. (https://arxiv.org/abs/1702.06516) 
[7] Erik G. Learned-Miller, John W. Fisher III. ICA using spacings estimates of entropy. Journal of Machine Learning Research, 4:1271-1295, 2003.
[8] Francis R. Bach. Michael I. Jordan. Kernel Independent Component Analysis. Journal of Machine Learning Research 3: 1-48, 2002.
[9] Hao Shen, Stefanie Jegelka and Arthur Gretton. Fast Kernel-Based Independent Component Analysis, IEEE Transactions on Signal Processing, 57:3498-3511, 2009.