Paper:
A new statistical procedure (Model-X Candès et al. (2018)) has provided a way to identify important factors using any supervised learning method controlling for FDR. This line of research has shown great potential to expand the horizon of machine learning methods beyond the task of prediction, to serve the broader needs in scientific researches for interpretable findings. However, the lack of a practical and flexible method to generate knockoffs remains the major obstacle for wide application of Model-X procedure. This paper fills in the gap by proposing a model-free knockoff generator which approximates the correlation structure between features through latent variable representation. We demonstrate our proposed method can achieve FDR control and better power than two existing methods in various simulated settings and a real data example for finding mutations associated with drug resistance in HIV-1 patients.

Review:
This manuscript tackles an important problem, namely, generating the knockoff procedure for FDR-controlled feature selection so that it can work on any data, rather than only for data generated from a Gaussian (as in the original work) or several specific other cases (mixtures of Gaussians or HMMs).  The basic idea is to exploit ideas from variational autoencoders to create a generic knockoff generation mechanism. Specifically, the main idea of the paper is to map the input covariates X into latent variable Z using a variational auto-encoder, generate the knockoffs \tilde{Z} in the latent space, and then map \tilde{Z} back to the input space to get the knockoffs \tilde{X}. The authors claim that their contribution in threefold: 
(1) Given the assumption is valid that X is mutually independent conditional on Z, the generated knockoffs \tilde{X} is proved to be valid in terms of satisfying the necessary swap condition. 
(2) Given (1) holds, and given that the discrepancy (measured by KL-divergence) between the true conditional probability Q(Z|X) and its estimate using variational auto-encoder is bounded by o((nlogp)^{-1/2}), the FDR is also bounded.
(3) The proposed knockoffs generating procedure can achieve controlled FDR and better power. 
In agreement with the above intuition, I have major concerns about the correctness of the paper.
The cornerstone of the proof in contribution (1) relies on the assumption that X is mutually independent conditional on Z. However, this assumption is invalid if there are dependencies between x_i and x_j. Therefore, the proposed knockoffs \tilde{X} cannot be proved valid by Theorem 2.1.
The erroneous proof in contribution (1) leads to the failure of contribution (2) stated in Theorem 2.2. The FDR is no longer controlled. Intuitively, according to algorithm 1, \tilde{Z} and \tilde{X} are expected to have the same distribution as Z and X, respectively; therefore, the FDR cannot be controlled.
The experimental results are suspicious. In general, it seems fishy that the proposed VAE approach outperforms Model X in the fully Gaussian case.  In this setting, Model X should have an advantage, since it is specifically designed for Gaussian generated data.  Related to this, the text is confusing: "Since the data were not Gaussian, the second-order matching method has the lowest power. The assumptions of the Fixed knockoff generations holds for the Gaussian cases, …" In the figure, the second-order matching method has low power even in the Gaussian case.
Minor comments:
p. 2: The exposition should explain the motivation for Knockoff+.
The manuscript contains numerous grammatical errors, a few examples of which are below:
p. 1: "biological linked" -> "biologically linked"
p. 1: "associated certain" -> "associated with a certain"
p. 1: "showed light" -> "shed light"
p. 1: "which has very limited" -> "which has limited"
p. 1: "leveraging on the power of of" -> "leveraging the power of"