Statistical RIP sampling matrices and robust recovery

Compressive sampling is a technique of recovering sparse N -dimensional signals from low-dimensional projections, i.e., their linear images in R,m ≪ N. In formal terms the problem can be stated as follows. Let Φ : R → R ,m ≪ N be a linear operator used to create a “sketch” of a signal represented by a real vector x ∈ R . In other words, we observe a compressed version of the signal, i.e., a vector r = Φx, where Φ is an m×N sampling matrix. Recovering x from r is generally impossible because the system of equations is under-determined, and the solutions form an affine subspace in R . The problem becomes tractable if we seek an approximation of x by a vector x̂ that satisfies