Reviews: Dimensionality reduction: theoretical perspective on practical measures

Originality As far as I can tell, the authors' claim that this is the first such work is correct. Previous work has been done is describing heuristics or empirical understandings of such behaviour, but the work is nonetheless original in proving a theoretical basis for this. Quality The authors' exposition of the problem and the solution is well thought out and expertly laid out in a logical and convincing form. However, the excellent technical contribution is somewhat lacking in discussion, particularly given the authors aim to bridge the gap between theory and practice; such claims as "This new consequence may serve an important guide for practical considerations" warrant a standalone discussion section which is not provided. Further, the results predicted in theory could have been compared to empirical experiments to show tightness in practice, and phase transitions could be shown in experiments as a demonstration.