A remark on the Restricted Isometry Property in Orthogonal Matching Pursuit

TL;DR

This paper establishes a precise condition on the restricted isometry constant under which Orthogonal Matching Pursuit guarantees exact recovery of sparse signals, confirming a conjecture from 2009.

Contribution

It proves a sharp bound on the restricted isometry constant ensuring OMP's success and constructs a counterexample showing the bound's tightness, confirming a prior conjecture.

Findings

OMP recovers all K-sparse signals if δ_{K+1} < 1/(√K+1)

A matrix with δ_{K+1} = 1/√K can fail OMP recovery

The result confirms a conjecture by Dai and Milenkovic (2009)

Abstract

This paper demonstrates that if the restricted isometry constant $\delta_{K+1}$ of the measurement matrix $A$ satisfies $$ \delta_{K+1} < \frac{1}{\sqrt{K}+1}, $$ then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover every $K$--sparse signal $\mathbf{x}$ in $K$ iterations from $A\x$. By contrast, a matrix is also constructed with the restricted isometry constant $$ \delta_{K+1} = \frac{1}{\sqrt{K}} $$ such that OMP can not recover some $K$-sparse signal $\mathbf{x}$ in $K$ iterations. This result positively verifies the conjecture given by Dai and Milenkovic in 2009.