Phase transitions phenomenon in Compressed Sensing

Compressed Sensing reconstruction algorithms typically exhibit a zeroth-order phase transition phenomenon for large problem sizes, where there is a domain of problem sizes for which successful recovery occurs with overwhelming probability, and there is a domain of problem sizes for which recovery failure occurs with overwhelming probability. The mathematics underlying this phenomenon will be outlined for $\ell1$ regularization and non-negative feasibility point regions. Both instances employ a large deviation analysis of the associated geometric probability event. These results give precise if and only if conditions on the number of samples needed in Compressed Sensing applications. Lower bounds on the phase transitions implied by the Restricted Isometry Property for Gaussian random matrices will also be presented for the following algorithms: $\ell^q$-regularization for $q\in (0,1]$, CoSaMP, Subspace Pursuit, and Iterated Hard Thresholding.