Reviews: Exact Recovery of Hard Thresholding Pursuit

This paper is quite technical but is very interesting and promising as it provides sufficient conditions for exact recovery or sparsistency of the solutions of an quite broad class sparsity constrained smooth optimization problems using a very simple, yet powerful, class of greedy algorithms. Despite the technicality of the results, they are very clearly synthesized and nicely presented. I have only a few (but important) concerns: Line 26: Problem (1) with a squared regression loss is IMO more general than compressed sensing. Depending on the forward model, it can tackle any noisy linear inverse problem such as optimal sparse representation in a dictionary, denoising, deblurring (or deconvolution),... Line 32: There is also that paper: [1]A. Line 81: least squared - least squares Line 121: In the definition of x {-*} the constraint with strict inequality leads to something that is ill-posed.