The authors systematically investigated the compressed sensing theory in the GWAS context. The paper was written clearly. I only have the following comments:

1. Although the empirical evaluation conducted in this paper show some interesting connections between CS and GWAS data analysis, it does not address the critical problems of GWAS in practice: (1) Can CS improve the power of identification of risk variants? (2) Can CS help to interpret GWAS results (about 85% of GWAS hits are in the non-coding region, Hindorff et al. (2009))? Could the author add some discussion about this?

L.A. Hindorff, P. Sethupathy, H.A. Junkins, E.M. Ramos, J.P. Mehta, F.S. Collins, and T.A. Manolio. Potential etiologic and functional implications of genome-wide association loci for human diseases and traits. Proceedings of the National Academy of Sciences, 106(23):9362, 2009.

1. The authors proposed to use median p-values of the selected nonzero coefficients as an observable measure. However, the so-call “p-value” is invalid because its calculation simply ignores the selection process of the sparse method. Significant tests in the sparse setting are still undergoing, e.g., Lockhart et al. (2013). The use of “p-value” could be misleading due to its invalidation.

Richard Lockhart, Jonathan Taylor, Ryan Tibshirani and Robert Tibshirani. (2013) A significance test for the lasso.

1. The choice of the tuning parameter lambda seems to be different from the standard cross-validation procedure. Could the author show some results based on cross-validation and compare to their current approach?

2. The author should add the single-variable-screening approach as a reference, in terms of FPR and PPV. It would be nice to demonstrate the advantage of CS over this single-variable-screening approach.

Level of interest: An article of importance in its field

Quality of written English: Acceptable

Statistical review: Yes, and I have assessed the statistics in my report.

Declaration of competing interests: I have no competing interests.

I am satisfied with most of the revision but I still would like to see the comparison between L1 and marginal regression in the context of GWAS. Because L1 is much more computationally demanding than Marginal regression, I would be highly appreciate that if we know what is the benefit.

Level of interest: An article of importance in its field

Quality of written English: Acceptable

Statistical review: Yes, and I have assessed the statistics in my report.

Declaration of competing interests: No conflict of interests.