Recht and Candes may champion approaches like compressed sensing, while Carlsson and Coifman align themselves more with the topological approach, but fundamenta

Recht and Candes may champion approaches like compressed sensing, while Carlsson and Coifman align themselves more with the topological approach, but fundamentally, these two methods are complementary rather than competitive. Minor nitpick, but I think the article (and this quote in particular) might be mischaracterizing the difference. Every form of manifold learning relies first and foremost on the ability to compute some kind of low-dimensional embedding that preserves structure, for which Compressed Sensing --and its more recent generalizations: see for instance the work of Wakin et al [1]-- clarifies the conditions and transformations that make them feasible. [1] http://papers.nips.cc/paper/3191-random-projections-for-manifold-learning.pdf http://papers.nips.cc/paper/3191-random-projections-for-mani...