Exact Sparse Signal Recovery via Orthogonal Matching Pursuit with Prior Information

The orthogonal matching pursuit (OMP) algorithm is a commonly used algorithm for recovering $K$-sparse signals $\x\in \mathbb{R}^{n}$ from linear model $\y=\A\x$, where $\A\in \mathbb{R}^{m\times n}$ is a sensing matrix. A fundamental question in the performance analysis of OMP is the characterization of the probability that it can exactly recover $\x$ for random matrix $\A$. Although in many practical applications, in addition to the sparsity, $\x$ usually also has some additional property (for example, the nonzero entries of $\x$ independently and identically follow the Gaussian distribution), none of existing analysis uses these properties to answer the above question. In this paper, we first show that the prior distribution information of $\x$ can be used to provide an upper bound on $\|\x\|_1^2/\|\x\|_2^2$, and then explore the bound to develop a better lower bound on the probability of exact recovery with OMP in $K$ iterations. Simulation tests are presented to illustrate the superiority of the new bound.