Sparsity and incoherence in orthogonal matching pursuit

Abstract

Recovery of sparse signals via approximation methods has been extensively studied in recently years. We consider the nonuniform recovery of orthogonal matching pursuit (OMP) from fewer noisy random measurements. Rows of sensing matrices are assumed to be drawn independently from a probability distribution obeying the isotropy property and the incoherence property. Our models not only include the standard sensing matrices in compressed sensing context, but also cover other new sensing matrices such as random convolutions, subsampled tight or continuous frames. Given m admissible random measurements of a fixed s-sparse signal $\varvec{x}\in \mathbb {R}^n$, we show that OMP can recover the support of $\varvec{x}$ exactly after s iterations with overwhelming probability provided that

$$\begin{aligned} m=O( s(s+\log (n-s))). \end{aligned}$$

It follows that the approximation order of OMP is

$$\begin{aligned} \Vert \varvec{x}- \varvec{x}_j\Vert =O(\eta ^j) \end{aligned}$$

where $0<\eta <1$ and $\varvec{x}_j$ denotes the recovered signal at j-th iteration. As a byproduct of the proof, the necessary number of measurements to ensure sparse recovery by $l_1$-minimization with random partial circulant or Toeplitz matrices is proved to be optimal.