This is a very interesting paper, which presents a comprehensive theoretical analysis of metric dimensionality reduction. It describe existing distortion measures in terms of moments of distortions and give an average case performance guarantee for these moments of distortion. Also, an approximate algorithm with provable guarantees on metric dimensionality reduction is introduced. The main objection on this paper was the absence of empirical evidence to support the claims. The authors have conducted additional experiments in the rebuttal phase but there are missing details regarding the experiments. The authors are advised to improve the quality of their paper in light of the reviewers' comments and incorporate their recommended changes.