Consider a measurement matrix A of size n x N, with n < N, y a signal in RN, 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 LQ-regularization with A with entries drawn i.i.d. from the Gaussian distribution N(0, 1 /√ N), 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 LQ-regularization will typically recover all k-sparse signals. Letting n/N → ∞, the aforementioned recoverability region is characterized by all ρ < (1 - ε)ρRIP(δ ; q) for any ε > 0, where ρRIP(δ ; q) is a lower bound of the true phase transition below which LQ-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 (Candès, 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.
Compressed sensing, sparsity, sparse approximation, sparse solutions to underdetermined systems, restricted isometry property, restricted isometry constants, phase transitions, convex relaxation, random matrices, Gaussian matrices, Wishart matrices, singular values of random matrices, eigenvalues of random matrices
Published in SIAM Review.