An Optimal Recovery Condition for Sparse Signals with Partial Support Information via OMP

Abstract

This paper considers the orthogonal matching pursuit (OMP) algorithm for sparse recovery in both noiseless and noisy cases when the partial prior information is available. The prior information is included in an estimated subset of the support of the sparse signal. First, we show that if $$\varvec{A}$$ satisfies $$\delta _{k+b+1}<\frac{1}{\sqrt{k-g+1}}$$ , then the OMP algorithm can perfectly recover any k-sparse signal $$\varvec{x}$$ from $$\varvec{y}=\varvec{Ax}$$ in $$k-g$$ iterations when the prior support of $$\varvec{x}$$ includes g true indices and b wrong indices. Furthermore, we show that the condition $$\delta _{k+b+1}<\frac{1}{\sqrt{k-g+1}}$$ is optimal. Second, we achieve the exact recovery of the remainder support (i.e., it is composed of indices in the true support of $$\varvec{x}$$ but not in the prior support) from $$\varvec{y}=\varvec{Ax}+\varvec{v}$$ under appropriate conditions. On the other hand, for the remainder support recovery, we also obtain a necessary condition based on the minimum magnitude of nonzero elements in the remainder support of $$\varvec{x}$$ . Compared to the OMP algorithm, numerical experiments demonstrate that the OMP algorithm with the partial prior information has better recovery performance.