Improved Bounds on the Restricted Isometry Constant for Orthogonal Matching Pursuit

TL;DR

This paper constructs counterexamples to show limits of the Orthogonal Matching Pursuit (OMP) algorithm's recovery capabilities based on the restricted isometry constant, and also identifies conditions under which OMP guarantees perfect recovery.

Contribution

The paper provides new bounds on the restricted isometry constant for OMP, including counterexamples and improved recovery guarantees, advancing understanding of OMP's performance limits.

Findings

Counterexamples for $ ext{RIP}$ constants where OMP fails

Conditions for perfect recovery with specific $ ext{RIP}$ bounds

Improvement over previous bounds by Mo et al. and Wang et al.

Abstract

In this letter, we first construct a counter example to show that for any given positive integer $K\geq 2$ and for any $\frac{1}{\sqrt{K+1}}\leq t<1$, there always exist a $K-$sparse $\x$ and a matrix $\A$ with the restricted isometry constant $\delta_{K+1}=t$ such that the OMP algorithm fails in $K$ iterations. Secondly, we show that even when $\delta_{K+1}=\frac{1}{\sqrt{K}+1}$, the OMP algorithm can also perfectly recover every $K-$sparse vector $\x$ from $\y=\A\x$ in $K$ iteration. This improves the best existing results which were independently given by Mo et al. and Wang et al.