Phase Transitions for Restricted Isometry Properties
Résumé
Currently there is no framework for the transparent comparison of sparse approximation recoverability results derived using different methods of analysis. We cast some of the most recent recoverability results for `1-regularization in terms of the phase transition framework advocated by Donoho. To allow for quantitative comparisons across different methods of analysis a particular random matrix ensemble must be selected; here we focus on Gaussian random matrices. Methods of analysis considered include the Restricted Isometry Property of Cand`es and Tao, geometric covering arguments of Rudelson and Vershynin, and convex polytopes formulations of Donoho.
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