A remark on the Restricted Isometry Property in Orthogonal Matching Pursuit

This paper demonstrates that if the restricted isometry constant $δ_{K+1}$ of the measurement matrix $A$ satisfies $δ_{K+1} < \frac{1}{\sqrt{K}+1},$ then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover every $K$--sparse signal $\mathbf{x}$ in $K$ iterations from $A\x$. By contrast, a matrix is also constructed with the restricted isometry constant $δ_{K+1} = \frac{1}{\sqrt{K}}$ such that OMP can not recover some $K$-sparse signal $\mathbf{x}$ in $K$ iterations. This result positively verifies the conjecture given by Dai and Milenkovic in 2009.

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