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.
The main difference between @cite_16 and our work is that we consider a more general completely perturbed scenario, and the optimality of the recovery performance is also in this sense. Compared with @cite_4 @cite_33 , the demand of RICs in our work is for the perturbed matrix @math other than @math , which is due to the fact that only @math is available for recovery. Also, the demand of RICs is with a constant parameter here. In addition, the condition such as ) or ) is not required in our assumption. Our results are compared with oracle recovery and shown to be optimal up to the coefficients, and they are verified by plentiful numerical simulations.
@cite_16 @cite_4 @cite_33
2039851473 1983749341 1561053739
This correspondence presents an average case denoising performance analysis for SP, CoSaMP, and IHT algorithms. This analysis considers the recovery of a noisy signal, with the assumptions that it is corrupted by an additive random zero-mean white Gaussian noise and has a K-sparse representation with respect to a known dictionary D . The proposed analysis is based on the RIP, establishing a near-oracle performance guarantee for each of these algorithms. Beyond bounds for the reconstruction error that hold with high probability, in this work we also provide a bound for the average error. Applications of compressed sensing motivate the possibility of using different operators to encode and decode a signal of interest. Since it is clear that the operators cannot be too different, we can view the discrepancy between the two matrices as a perturbation. The stability of l 1 -minimization and greedy algorithms to recover the signal in the presence of additive noise is by now well-known. Recently however, work has been done to analyze these methods with noise in the measurement matrix, which generates a multiplicative noise term. This new framework of generalized perturbations (i.e., both additive and multiplicative noise) extends the prior work on stable signal recovery from incomplete and inaccurate measurements of Candes, Romberg and Tao using Basis Pursuit (BP), and of Needell and Tropp using Compressive Sampling Matching Pursuit (CoSaMP). We show, under reasonable assumptions, that the stability of the reconstructed signal by both BP and CoSaMP is limited by the noise level in the observation. Our analysis extends easily to arbitrary greedy methods. In this paper, the Subspace Pursuit (SP) recovery of signals with sensing matrix perturbations is analyzed. Previous studies have only considered the robustness of Basis pursuit and greedy algorithms to recover the signal in the presence of additive noise with measurement and or signal. Since it is impractical to exactly implement the sampling matrix A in a physical sensor, precision errors must be considered. Recently, work has been done to analyze the methods with noise in the sampling matrix, which generates a multiplicative noise term. This new perturbed framework (both additive and multiplicative noise) extends the prior work of Basis pursuit and greedy algorithms on stable signal recovery from incomplete and inaccurate measurements. Our works show that, under reasonable conditions, the stability of the SP solution of the completely perturbed scenario was limited by the total noise in the observation.