On the Theoretical Analysis of Orthogonal Matching Pursuit with Termination Based on the Residue

Orthogonal Matching Pursuit (OMP) is a simple, yet empirically competitive algorithm for sparse recovery. Recent developments have shown that OMP guarantees exact recovery of $K$-sparse signals in $K$ iterations if the observation matrix $\Phi$ satisfies the Restricted Isometry Property (RIP) with Restricted Isometry Constant (RIC) $\delta_{K+1}<\frac{1}{\sqrt{K}+1}$. On the other hand, OMP empirically promises higher recovery rates when it runs for more than $K$ iterations. In order to support this theoretically, we extend the theoretical analysis of OMP to cover more than $K$ iterations. We develop exact recovery guarantees for $K$-sparse signals in more than $K$ iterations when $\Phi$ satisfies an RIP condition which depends on the number of correct and false indices in the support estimates of intermediate iterations. In addition, we present an upper bound on the number of false indices in the support estimate for the derived RIP condition to be less restrictive than $\delta_{K+1}<\frac{1}{\sqrt{K}+1}$. Moreover, we provide recovery simulations which demonstrate the performance improvement when more than $K$ iterations are allowed. Finally, we empirically analyse the number of false indices in the support estimate, which indicates that these do not violate the developed upper bound in practice.