A remark about orthogonal matching pursuit algorithm

TL;DR

This paper analyzes the theoretical performance of the Orthogonal Matching Pursuit (OMP) algorithm in compressed sensing, demonstrating its instance optimality under various conditions and distributions.

Contribution

It establishes new theoretical guarantees for OMP, including $(p,q)$ instance optimality for a broad class of encoders and probabilistic $(2,2)$ optimality for certain random matrices.

Findings

OMP achieves $(p,q)$ instance optimality for many encoders.

OMP is $(2,2)$ instance optimal in probability with certain random matrices.

Theoretical insights improve understanding of OMP's performance in compressed sensing.

Abstract

In this note, we investigate the theoretical properties of Orthogonal Matching Pursuit (OMP), a class of decoder to recover sparse signal in compressed sensing. In particular, we show that the OMP decoder can give $(p,q)$ instance optimality for a large class of encoders with $1\leq p\leq q \leq 2$ and $(p,q)\neq (2,2)$. We also show that, if the encoding matrix is drawn from an appropriate distribution, then the OMP decoder is $(2,2)$ instance optimal in probability.