After a careful discussion among the reviewers, there is a clear consensus that the paper provides a solid theoretical contributions for kernel density estimation in high dimensions. Hence, I am happy to recommend to accept this paper for publication at NeurIPS2019. Nevertheless, one concern that came up during the discussion is the lack of experimental comparison and clarity in how it was performed. I urge the authors to incorporate reviewers' comment on this weakness of the paper into the camera-ready version.