Coherence-Based Performance Guarantees of Orthogonal Matching Pursuit

TL;DR

This paper establishes coherence-based performance guarantees for Orthogonal Matching Pursuit (OMP) in noisy settings, showing that under strong coherence conditions and sufficient measurements, OMP reliably recovers sparse signals with high probability.

Contribution

It provides new coherence-based theoretical guarantees for OMP's support recovery and signal reconstruction, distinguishing matrix properties from signal properties.

Findings

OMP recovers sparse signals with high probability under strong coherence.

Performance guarantees depend on minimum SNR rather than signal power.

Analysis covers variants with known sparsity and stopping rules.

Abstract

In this paper, we present coherence-based performance guarantees of Orthogonal Matching Pursuit (OMP) for both support recovery and signal reconstruction of sparse signals when the measurements are corrupted by noise. In particular, two variants of OMP either with known sparsity level or with a stopping rule are analyzed. It is shown that if the measurement matrix $X\in\mathbb{C}^{n\times p}$ satisfies the strong coherence property, then with $n\gtrsim\mathcal{O}(k\log p)$, OMP will recover a $k$-sparse signal with high probability. In particular, the performance guarantees obtained here separate the properties required of the measurement matrix from the properties required of the signal, which depends critically on the minimum signal to noise ratio rather than the power profiles of the signal. We also provide performance guarantees for partial support recovery. Comparisons are given with other performance guarantees for OMP using worst-case analysis and the sorted one step thresholding algorithm.