A Sharp Restricted Isometry Constant Bound of Orthogonal Matching Pursuit

We shall show that if the restricted isometry constant (RIC) $\delta_{s+1}(A)$ of the measurement matrix $A$ satisfies $$\delta_{s+1}(A) < \frac{1}{\sqrt{s + 1}},$$ 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 $s \in \N$, we shall construct a matrix $A$ with the RIC $$\delta_{s+1}(A) = \frac{1}{\sqrt{s + 1}}$$ such that OMP may not recover some $s$-sparse signal $x$ in $s$ iterations.