A Sharp Restricted Isometry Constant Bound of Orthogonal Matching Pursuit

TL;DR

This paper establishes a sharp bound on the restricted isometry constant for the success of Orthogonal Matching Pursuit in recovering sparse signals, and demonstrates the bound's optimality with a counterexample.

Contribution

It provides a precise RIC bound for OMP success and proves this bound is tight through a constructed counterexample.

Findings

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

The RIC bound is proven to be sharp with a constructed matrix example

Recovery failure occurs at the bound δ_{s+1}(A) = 1/√(s+1)

Abstract

We shall show that if the restricted isometry constant (RIC) $\delta_{s+1}(A)$ of the measurement matrix $A$ satisfies $$ \delta_{s+1}(A) < \frac{1}{\sqrt{s + 1}}, $$ then the greedy algorithm Orthogonal Matching Pursuit(OMP) will succeed. That is, OMP can recover every $s$-sparse signal $x$ in $s$ iterations from $b = Ax$. Moreover, we shall show the upper bound of RIC is sharp in the following sense. For any given $s \in \N$, we shall construct a matrix $A$ with the RIC $$ \delta_{s+1}(A) = \frac{1}{\sqrt{s + 1}} $$ such that OMP may not recover some $s$-sparse signal $x$ in $s$ iterations.