[Paper Review] Sharp Conditions for Exact Support Recovery of Sparse Signals with Noise via OMP.

Jinming Wen, Zhengchun Zhou+3 more|arXiv (Cornell University)|Dec 22, 2015

Sparse and Compressive Sensing Techniques24 references3 citations

TL;DR

This paper establishes sharp sufficient and necessary conditions for exact support recovery of sparse signals with noise using Orthogonal Matching Pursuit (OMP). It shows that when the restricted isometry constant $\delta_{K+1}(\mathbf{A}) < \frac{1}{\sqrt{K+1}}$ and the minimum magnitude of nonzero signal components meets mild constraints, OMP exactly recovers the true support under $l_2$ and $l_\infty$ bounded noise, with the bound being optimal in terms of the RIC.

ABSTRACT

Sufficient conditions for exact support recovery of a sparse signal from noisy measurements with orthogonal matching pursuit (OMP) have been extensively studied in the literature. In this paper, we first show that if the restricted isometry constant (RIC) $\delta_{k+1}(\A)$ of the sensing matrix $\A$ satisfies $\delta_{K+1}(\A) < \frac{1}{\sqrt {K+1} }$, then under some constraints on the minimum magnitude of the nonzero elements of the $K-$sparse signal $\x$, the support of $\x$ can be exactly recovered from the measurements $\y=\A\x+\v$ by OMP under the $l_2$ and $l_{\infty}$ bounded noises. These two conditions are sharp in terms of the RIC since 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 $\A$ satisfying $\delta_{K+1}(\A)=t$ for which OMP may fail to recover the signal $\x$. Our constraints on the minimum magnitude of nonzero elements of $\x$ are also much weaker than existing ones. Moreover, we propose some necessary conditions for the exact support recovery of $\x$ on the minimum magnitude of the nonzero elements of $\x$.

Motivation & Objective

• To establish tight sufficient conditions for exact support recovery of $K$-sparse signals using OMP under noisy measurements.
• To identify the weakest possible constraints on the minimum magnitude of nonzero signal components for successful recovery.
• To provide necessary conditions on signal magnitude for exact support recovery, completing the theoretical characterization.
• To demonstrate the sharpness of the RIC threshold $\delta_{K+1} < \frac{1}{\sqrt{K+1}}$ by constructing counterexamples for any larger threshold.

Proposed method

• Derivation of sufficient conditions for exact support recovery based on the restricted isometry constant $\delta_{K+1}(\mathbf{A})$ of the sensing matrix $\mathbf{A}$.
• Analysis of OMP performance under $l_2$ and $l_\infty$ bounded noise models.
• Introduction of a new, weaker lower bound on the minimum magnitude of nonzero signal components compared to prior work.
• Construction of explicit counterexamples for any $\delta_{K+1} = t \geq \frac{1}{\sqrt{K+1}}$, proving the sharpness of the RIC condition.
• Derivation of necessary conditions on the signal's minimum nonzero component magnitude for exact recovery.

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound on the restricted isometry constant $\delta_{K+1}(\mathbf{A})$ that guarantees exact support recovery of a $K$-sparse signal via OMP under noisy conditions?
• RQ2 How weak can the minimum magnitude of the nonzero components of a sparse signal be while still allowing exact support recovery by OMP?
• RQ3 Can the sufficient condition $\delta_{K+1}(\mathbf{A}) < \frac{1}{\sqrt{K+1}}$ be shown to be sharp, i.e., is failure possible when this bound is exceeded?
• RQ4 What are the necessary conditions on the minimum magnitude of nonzero signal components for OMP to succeed in support recovery?
• RQ5 How do the proposed conditions compare in strength to existing results in terms of signal magnitude constraints?

Key findings

• The condition $\delta_{K+1}(\mathbf{A}) < \frac{1}{\sqrt{K+1}}$ is sufficient for exact support recovery of $K$-sparse signals under $l_2$ and $l_\infty$ bounded noise, provided the minimum magnitude of nonzero signal components satisfies a mild constraint.
• The proposed constraint on the minimum magnitude of nonzero components is significantly weaker than those in existing literature.
• For any $K \geq 2$ and any $t \in \left[\frac{1}{\sqrt{K+1}}, 1\right)$, there exist a $K$-sparse signal $\mathbf{x}$ and a sensing matrix $\mathbf{A}$ with $\delta_{K+1}(\mathbf{A}) = t$ such that OMP fails to recover the support, proving the sharpness of the RIC threshold.
• The paper derives necessary conditions on the minimum magnitude of nonzero signal components for exact recovery, completing the theoretical picture.
• The results show that the RIC threshold $\frac{1}{\sqrt{K+1}}$ is optimal and cannot be relaxed without risking failure in support recovery.