Oracle-order Recovery Performance of Greedy Pursuits with Replacement against General Perturbations

TL;DR

This paper analyzes the stability and performance of greedy pursuit algorithms with replacement, such as CoSaMP, SP, and IHT, in sparse recovery under general measurement and sensing matrix perturbations, demonstrating oracle-order recovery guarantees.

Contribution

It provides a unified error bound analysis for greedy pursuits with replacement under perturbations, revealing their oracle-order recovery performance in practical noisy conditions.

Findings

Recovery algorithms are stable against perturbations.

Error bounds are close to oracle recovery bounds.

Numerical simulations confirm theoretical results.

Abstract

Applying the theory of compressive sensing in practice always takes different kinds of perturbations into consideration. In this paper, the recovery performance of greedy pursuits with replacement for sparse recovery is analyzed when both the measurement vector and the sensing matrix are contaminated with additive perturbations. Specifically, greedy pursuits with replacement include three algorithms, compressive sampling matching pursuit (CoSaMP), subspace pursuit (SP), and iterative hard thresholding (IHT), where the support estimation is evaluated and updated in each iteration. Based on restricted isometry property, a unified form of the error bounds of these recovery algorithms is derived under general perturbations for compressible signals. The results reveal that the recovery performance is stable against both perturbations. In addition, these bounds are compared with that of oracle recovery--- least squares solution with the locations of some largest entries in magnitude known a priori. The comparison shows that the error bounds of these algorithms only differ in coefficients from the lower bound of oracle recovery for some certain signal and perturbations, as reveals that oracle-order recovery performance of greedy pursuits with replacement is guaranteed. Numerical simulations are performed to verify the conclusions.