Consider a measurement matrix A of size n×N, with n < N, y a signal in R^N, and b = Ay the observed measurement of the vector y. From knowledge of (b,A), compressed sensing seeks to recover the k-sparse x, k < n, which minimizes ||b-Ax||. Using various methods of analysis — convex polytopes, geometric functional analysis, and the restricted isometry property (RIP) — it has been proven that x can be reconstructed via l^q-regularization (q 2 (0, 1]) provided A satisfies conditions dictated by the method of analysis. This article focuses on the RIP approach and A with entries drawn i.i.d. from the Gaussian distribution, developing more precise bounds on the restricted isometry constants, and using these bounds in an asymmetric RIP formulation to quantify the region of (n/N , k/n) in which RIP implies that l^q-regularization will typically recover all k-sparse signals. Letting n/N -->\delta and k/n --> \rho as n grows to infinity, the aforementioned recoverability region is characterized by all \rho< (1-\epsilon)\rho_S for any \epsilon > 0, where \rho_S is a lower bound of the true phase transition below which l^q-regularization will typically recover all k-sparse signals. This phase transition framework, proposed in this context by Donoho (2005), is applied to compressed sensing results obtained by the analysis techniques of centro-symmetric polytope theory (Donoho), geometric functional analysis (Rudelson and Vershynin), and the RIP (Candes, Romberg and Tao; Foucart and Lai; Chartrand). Recasting the results from different methods of analysis into a common phase transition framework allows for the direct comparison of the efficacy of the respective results.
Technical Report, School of Mathematics, University of Edinburgh, 2009