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
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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. This paper considers the problem of recovering an unknown sparse p×p matrix X from an m×m matrix Y=AXB T , where A and B are known m×p matrices with m≪p. The main result shows that there exist constructions of the sketching matrices A and B so that even if X has O(p) nonzeros, it can be recovered exactly and efficiently using a convex program as long as these nonzeros are not concentrated in any single row column of X. Furthermore, it suffices for the size of Y (the sketch dimension) to scale as m = O(√(# nonzeros in X) × log p). The results also show that the recovery is robust and stable in the sense that if X is equal to a sparse matrix plus a perturbation, then the convex program we propose produces an approximation with accuracy proportional to the size of the perturbation. Unlike traditional results on sparse recovery, where the sensing matrix produces independent measurements, our sensing operator is highly constrained (it assumes a tensor product structure). Therefore, proving recovery guarantees require nonstandard techniques. Indeed, our approach relies on a novel result concerning tensor products of bipartite graphs, which may be of independent interest. This problem is motivated by the following application, among others. Consider a p×n data matrix D, consisting of n observations of p variables. Assume that the correlation matrix X:=DD T is (approximately) sparse in the sense that each of the p variables is significantly correlated with only a few others. Our results show that these significant correlations can be detected even if we have access to only a sketch of the data S=AD with A ∈ R m×p . Motivated by the recent developments of nonconvex penalties in sparsity modeling, we propose a nonconvex optimization model for handing the low-rank matrix recovery problem. Different from the famous robust principal component analysis (RPCA), we suggest recovering low-rank and sparse matrices via a nonconvex loss function and a nonconvex penalty. The advantage of the nonconvex approach lies in its stronger robustness. To solve the model, we devise a majorization-minimization augmented Lagrange multiplier (MM-ALM) algorithm which finds the local optimal solutions of the proposed nonconvex model. We also provide an efficient strategy to speedup MM-ALM, which makes the running time comparable with the state-of-the-art algorithm of solving RPCA. Finally, empirical results demonstrate the superiority of our nonconvex approach over RPCA in terms of matrix recovery accuracy. The current parallel storage systems use thousands of inexpensive disks to meet the storage requirement of applications. Data redundancy and or coding are used to enhance data availability, for instance, Row-diagonal parity (RDP) and EVENODD codes, which are widely used in RAID-6 storage systems, provide data availability with up to two disk failures. To reduce the probability of data unavailability, whenever a single disk fails, disk recovery will be carried out. We find that the conventional recovery schemes of RDP and EVENODD codes for a single failed disk only use one parity disk. However, there are two parity disks in the system, and both can be used for single disk failure recovery. In this article, we propose a hybrid recovery approach that uses both parities for single disk failure recovery, and we design efficient recovery schemes for RDP code (RDOR-RDP) and EVENODD code (RDOR-EVENODD). Our recovery scheme has the following attractive properties: (1) “read optimality” in the sense that our scheme issues the smallest number of disk reads to recover a single failed disk and it reduces approximately 1 4 of disk reads compared with conventional schemes; (2) “load balancing property” in that all surviving disks will be subjected to the same (or almost the same) amount of additional workload in rebuilding the failed disk. We carry out performance evaluation to quantify the merits of RDOR-RDP and RDOR-EVENODD on some widely used disks with DiskSim. The offline experimental results show that RDOR-RDP and RDOR-EVENODD outperform the conventional recovery schemes of RDP and EVENODD codes in terms of total recovery time and recovery workload on individual surviving disk. However, the improvements are less than the theoretical value (approximately 25p), as RDOR-RDP and RDOR-EVENODD change the disk access pattern from purely sequential to a more random one compared with their conventional schemes.