A Sharp Restricted Isometry Constant Bound of Orthogonal Matching Pursuit

We shall show that if the restricted isometry constant (RIC) δs+1(A) of the measurement matrix A satisfies δs+1(A) < 1 √ s+ 1 , then the greedy algorithm Orthogonal Matching Pursuit(OMP) will succeed. That is, OMP can recover every s-sparse signal x in s iterations from b = Ax. Moreover, we shall show the upper bound of RIC is sharp in the following sense. For any given s ∈ N, we shall construct a matrix A with the RIC δs+1(A) = 1 √ s+ 1 such that OMP may not recover some s-sparse signal x in s iterations. Index Terms Compressed sensing, restricted isometry property, orthogonal matching pursuit, sparse signal reconstruction.