On the Noise Robustness of Simultaneous Orthogonal Matching Pursuit

TL;DR

This paper analyzes the robustness of simultaneous orthogonal matching pursuit (SOMP) in recovering joint support of multiple noisy sparse signals, revealing limitations when the number of signals grows large.

Contribution

It introduces a novel upper bounding analysis of SOMP's probability of incorrect support recovery under noisy conditions with multiple signals.

Findings

SOMP's error probability can be bounded in noisy scenarios.

As the number of signals increases, error probability may not vanish even in noiseless cases.

Simulation results validate the theoretical bounds and insights.

Abstract

In this paper, the joint support recovery of several sparse signals whose supports present similarities is examined. Each sparse signal is acquired using the same noisy linear measurement process, which returns fewer observations than the dimension of the sparse signals. The measurement noise is assumed additive, Gaussian, and admits different variances for each sparse signal that is measured. Using the theory of compressed sensing, the performance of simultaneous orthogonal matching pursuit (SOMP) is analysed for the envisioned signal model. The cornerstone of this paper is a novel analysis method upper bounding the probability that SOMP recovers at least one incorrect entry of the joint support during a prescribed number of iterations. Furthermore, the probability of SOMP failing is investigated whenever the number of sparse signals being recovered simultaneously increases and tends to infinity. In particular, convincing observations and theoretical results suggest that SOMP committing no mistake in the noiseless case does not guarantee the absence of error in the noisy case whenever the number of acquired sparse signals scales to infinity. Finally, simulation results confirm the validity of the theoretical results.