Paper:
Unsupervised learning is about capturing dependencies between variables and driven by the contrast between the probable vs improbable configurations of these variables, often either via a generative model which only samples probable ones or with an energy function (unnormalized log-density) which is low for probable ones and high for improbable ones. Here we consider learning both an energy function and an efficient approximate sampling mechanism for the corresponding distribution. Whereas the critic (or discriminator) in generative adversarial networks (GANs) learns to separate data and generator samples, introducing an entropy maximization regularizer on the generator can turn the interpretation of the critic into an energy function, which separates the training distribution from everything else, and thus can be used for tasks like anomaly or novelty detection. This paper is motivated by the older idea of sampling in latent space rather than data space because running a Monte-Carlo Markov Chain (MCMC) in latent space has been found to be easier and more efficient, and because a GAN-like generator can convert latent space samples to data space samples. For this purpose, we show how a Markov chain can be run in latent space whose samples can be mapped to data space, producing better samples. These samples are also used for the negative phase gradient required to estimate the log-likelihood gradient of the data space energy function. To maximize entropy at the output of the generator, we take advantage of recently introduced neural estimators of mutual information. We find that in addition to producing a useful scoring function for anomaly detection, the resulting approach produces sharp samples (like GANs) while covering the modes well, leading to high Inception and Frchet scores.

Review:
Thank you for an interesting read.
The paper proposes an approximate training technique for energy-based models (EBMs). More specifically, the samples used negative phase gradient in EBM training is approximated by samples from another generator. This "approximate generator" is a composition of a decoder (which, with a Gaussian prior on latent variable z, is trained to approximate the data distribution) and another EBM in latent space. The authors show connections to WGAN training, thus the name EnGAN. Experiments on natural image generation and anomaly detection show promising improvements, although not very significant.
From my understanding of the paper, the main contribution of the paper comes from section 4, which proposes a latent-space MCMC scheme to improve sample quality. I have seen several papers fusing EBMs and GAN training together and to the best of my knowledge section 4 is novel (but with problems, see below). Section 3's recipe is quite standard, e.g. as seen in Kim and Bengio (2017), and in principle contrastive divergence also uses the same idea. The idea of estimating of the entropy term for the implicit distribution p_G with adversarial mutual information estimation is something new, although quite straight-forward.
Although I do agree that MCMC mixing in x space can be much harder than MCMC mixing in z space, since I don't think the proposed latent-space MCMC scheme is exact (apart from finite-time simulation, rejection...), I don't see theoretically why the method works.
1. The MCMC method essentially samples z from another EBM, where that EBM(z) has energy function -E_{\theta}(G(z)), and then generate x = G(z). Note here EBM(z) != p(z). The key issue is, even when p_G(x) = p_{\theta}(x), there is no guarantee that the proposed latent-space MCMC method would return x samples according to distribution p_{\theta}(x). You can easily work out a counter example by considering G is an invertible transformation. Therefore I don't understand why doing MCMC on this latent-space EBM can help improve sample quality in x space.
2. Continuing point 1, with Algorithm 1 that only fits p_G(x) towards p_{\theta}(x), I am confident that the negative phase gradient is still quite biased. Why not just use the latent-space MCMC sampler composited with G as the generator, and use these MCMC samples to train both the decoder G and the mutual information estimator?
3. I am not exactly sure why the gradient norm regulariser in (3) make sense here? True that it would be helpful to correct the bias of the negative phase, but why this particular form? We are not doing WGAN here and in general we don't usually put a Lipschitz constraint on the energy function. I've seem several GAN papers arguing that gradient penalty helps in cases beyond WGAN, but most of them are just empirical observations...
Also the Omega regulariser is computed on which x? On data? Do you know whether the energy is guaranteed to be minimized at data locations? In this is that appropriate to call Omega a regulariser?
The presentation is overall clear, although I think there are a few typos and confusing equations:
1. There should be a negative sign on the LHS of equation 2.
2. Equation 3 is inconsistent with the energy update equation in Algorithm 1. The latter one makes more sense.
3. Where is the ratio between the transition kernels in the acceptance ratio equation? In general for Langevin dynamics the transition kernel is not symmetric.