The maximum mean discrepancy (MMD) between two probability measures P and Q is a metric that is zero if and only if all moments of the two measures are equal, making it an appealing statistic for two-sample tests. Given i.i.d. samples from P and Q, Gretton et al. (2012) show that we can construct an unbiased estimator for the square of the MMD between the two distributions. If P is a distribution of interest and Q is the distribution implied by a generative neural network with stochastic inputs, we can use this estimator to train our neural network. However, in practice we do not always have i.i.d. samples from our target of interest. Data sets often exhibit biases-for example, under-representation of certain demographics-and if we ignore this fact our machine learning algorithms will propagate these biases. Alternatively, it may be useful to assume our data has been gathered via a biased sample selection mechanism in order to manipulate properties of the estimating distribution Q.
In this paper, we construct an estimator for the MMD between P and Q when we only have access to P via some biased sample selection mechanism, and suggest methods for estimating this sample selection mechanism when it is not already known. We show that this estimator can be used to train generative neural networks on a biased data sample, to give a simulator that reverses the effect of that bias.
This paper proposes an importance-weighted estimator of the MMD, in order to estimate the MMD between distributions based on samples biased according to a known scheme. It then discusses how to estimate the scheme when it is unknown, and further proposes using it in either the MMD-based generative models of Y. Li et al. (2015) / Dziugaite et al. (2015), or in the MMD GAN of C.-L. Li et al. (2017).
The estimator itself is natural (and relatively obvious), though it has some drawbacks that aren't fully discussed (below).
The application to GAN-type learning is reasonable, and topical. The first, univariate, experiment shows that the scheme is at least plausible. But the second experiment, involving a simple T ratio based on whether an MNIST digit is a 0 or a 1, doesn't even really work! (The best model only gets the underrepresented class from 20% up to less than 40%, rather than the desired 50%, and the "more realistic" setting only to 33%.)
It would be helpful to debug whether this is due to the classifier being incorrect, estimator inaccuracies, or what. In particular, I would try using T based on a pretrained convnet independent of the autoencoder representation in the MMD GAN, to help diagnose where the failure mode comes from.
Without at least a working should-be-easy example like this, and with the rest of the paper's technical contribution so small, I just don't think this paper is ready for ICLR.
It's also worth noting that the equivalent algorithm for either vanilla GANs or Wasserstein GANs would be equally obvious.
Estimator:
In the discussion about (2): where does the 1/m bias come from? This doesn't seem to be in Robert and Casella section 3.3.2, which is the part of the book that I assume you're referring to (incidentally, you should specify that rather than just citing a 600-page textbook).
Moreover, it is worth noting that Robert and Cassela emphasize that if E[1 / \tilde T] is infinite, the importance sampling estimator can be quite bad (for example, the estimator may have infinite variance). This happens when \tilde T puts mass in a neighborhood around 0, i.e. when the thinned distribution doesn't have support at any place that P does. In the biased-observations case, this is in some sense unsurprising: if we don't see *any* data in a particular class of inputs, then our estimates can be quite bad (since we know nothing about a group of inputs that might strongly affect the results). In the modulating case, the equivalent situation is when F(x) lacks a mean, which seems less likely. Thus although this is probably not a huge problem for your case, it's worth at least mentioning. (See also the following relevant blog posts:  https://radfordneal.wordpress.com/2008/08/17/the-harmonic-mean-of-the-likelihood-worst-monte-carlo-method-ever/ and https://xianblog.wordpress.com/2012/03/12/is-vs-self-normalised-is/ .)
The paper might be improved by stating (and proving) a theorem with expressions for the rate of convergence of the estimator, and how they depend on T.
Minor:
Another piece of somewhat-related work is Xiong and Schneider, Learning from Point Sets with Observational Bias, UAI 2014.
Sutherland et al. 2016 and 2017, often referenced in the same block of citations, are the same paper.
On page 3, above (1): "Since we have projected the distributons into an infinite-dimensional space, the distance between the two distributions is zero if and only if all their moments are the same." An infinite-dimensional space isn't enough; the kernel must further be characteristic, as you mention. See e.g. Sriperumbuder et al. (AISTATS 2010) for more details.
Figure 1(b) seems to be plotting only the first term of \tilde T, without the + 0.5.