Signal-Dependent Performance Analysis of Orthogonal Matching Pursuit for Exact Sparse Recovery

Exact recovery of $K$-sparse signals $x \in \mathbb{R}^{n}$ from linear measurements $y=Ax$, where $A\in \mathbb{R}^{m\times n}$ is a sensing matrix, arises from many applications. The orthogonal matching pursuit (OMP) algorithm is widely used for reconstructing $x$. A fundamental question in the performance analysis of OMP is the characterizations of the probability of exact recovery of $x$ for random matrix $A$ and the minimal $m$ to guarantee a target recovery performance. In many practical applications, in addition to sparsity, $x$ also has some additional properties. This paper shows that these properties can be used to refine the answer to the above question. In this paper, we first show that the prior information of the nonzero entries of $x$ can be used to provide an upper bound on $\|x\|_1^2/\|x\|_2^2$. Then, we use this upper bound to develop a lower bound on the probability of exact recovery of $x$ using OMP in $K$ iterations. Furthermore, we develop a lower bound on the number of measurements $m$ to guarantee that the exact recovery probability using $K$ iterations of OMP is no smaller than a given target probability. Finally, we show that when $K=O(\sqrt{\ln n})$, as both $n$ and $K$ go to infinity, for any $0<\zeta\leq 1/\sqrt{\pi}$, $m=2K\ln (n/\zeta)$ measurements are sufficient to ensure that the probability of exact recovering any $K$-sparse $x$ is no lower than $1-\zeta$ with $K$ iterations of OMP. For $K$-sparse $\alpha$-strongly decaying signals and for $K$-sparse $x$ whose nonzero entries independently and identically follow the Gaussian distribution, the number of measurements sufficient for exact recovery with probability no lower than $1-\zeta$ reduces further to $m=(\sqrt{K}+4\sqrt{\frac{\alpha+1}{\alpha-1}\ln(n/\zeta)})^2$ and asymptotically $m\approx 1.9K\ln (n/\zeta)$, respectively.