A sharp bound on RIC in generalized orthogonal matching pursuit

TL;DR

This paper establishes a sharp bound on the restricted isometry constant for the generalized orthogonal matching pursuit algorithm, ensuring exact sparse signal recovery in noiseless and noisy scenarios, and demonstrates the bound's optimality.

Contribution

It provides the first sharp RIC bound for gOMP's exact recovery of sparse signals and constructs matrices that show the bound's tightness.

Findings

The RIC bound for perfect recovery is _{NK+1}<1/(rac{K}{N}+1).

The bound is proven to be sharp via explicit matrix construction.

In noisy cases, additional conditions on signal magnitude enable support recovery.

Abstract

Generalized orthogonal matching pursuit (gOMP) algorithm has received much attention in recent years as a natural extension of orthogonal matching pursuit. It is used to recover sparse signals in compressive sensing. In this paper, a new bound is obtained for the exact reconstruction of every $K$-sparse signal via the gOMP algorithm in the noiseless case. That is, if the restricted isometry constant (RIC) $\delta_{NK+1}$ of the sensing matrix $A$ satisfies \begin{eqnarray*} \delta_{NK+1}<\frac{1}{\sqrt{\frac{K}{N}+1}}, \end{eqnarray*} then the gOMP can perfectly recover every $K$-sparse signal $x$ from $y=Ax$. Furthermore, the bound is proved to be sharp in the following sense. For any given positive integer $K$, we construct a matrix $A$ with the RIC \begin{eqnarray*} \delta_{NK+1}=\frac{1}{\sqrt{\frac{K}{N}+1}} \end{eqnarray*} such that the gOMP may fail to recover some $K$-sparse signal $x$. In the noise case, an extra condition on the minimum magnitude of the nonzero components of every $K-$sparse signal combining with the above bound on RIC of the sensing matrix $A$ is sufficient to recover the true support of every $K$-sparse signal by the gOMP.