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.
In other relevant works @cite_4 @cite_33 , the error bounds of CoSaMP and SP are also derived under general perturbations. The analysis is performed following the steps of @cite_3 for basis pursuit. Define The main result in @cite_4 states that under the conditions that and the recovered solution of CoSaMP satisfies Further define where The main result in @cite_33 states that under the conditions that and that the recovered solution of SP satisfies
@cite_4 @cite_33 @cite_3
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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. We analyze the Basis Pursuit recovery of signals with general perturbations. Previous studies have only considered partially perturbed observations Ax + e. Here, x is a signal which we wish to recover, A is a full-rank matrix with more columns than rows, and e is simple additive noise. Our model also incorporates perturbations E to the matrix A which result in multiplicative noise. This completely perturbed framework extends the prior work of Candes, Romberg, and Tao on stable signal recovery from incomplete and inaccurate measurements. Our results show that, under suitable conditions, the stability of the recovered signal is limited by the noise level in the observation. Moreover, this accuracy is within a constant multiple of the best-case reconstruction using the technique of least squares. In the absence of additive noise, numerical simulations essentially confirm that this error is a linear function of the relative perturbation.