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This paper is concerned with the problem of recovering a structured signal from a relatively small number of corrupted random measurements. Sharp phase transitions have been numerically observed in practice when different convex programming procedures are used to solve this problem. This paper is devoted to presenting theoretical explanations for these phenomena by employing some basic tools from Gaussian process theory. Specifically, we identify the precise locations of the phase transitions for both constrained and penalized recovery procedures. Our theoretical results show that these phase transitions are determined by some geometric measures of structure, e.g., the spherical Gaussian width of a tangent cone and the Gaussian (squared) distance to a scaled subdifferential. By utilizing the established phase transition theory, we further investigate the relationship between these two kinds of recovery procedures, which also reveals an optimal strategy (in the sense of Lagrange theory) for choosing the tradeoff parameter in the penalized recovery procedure. Numerical experiments are provided to verify our theoretical results. [ABSTRACT FROM AUTHOR]

Corrupted sensing concerns the problem of recovering a high-dimensional structured signal from a collection of measurements that are contaminated by unknown structured corruption and unstructured noise. In the case of linear measurements, the recovery performance of different convex programming procedures (e.g., generalized Lasso and its variants) is well established in the literature. However, in practical applications of digital signal processing, the quantization process is inevitable, which often leads to non-linear measurements. This paper is devoted to studying corrupted sensing under quantized measurements. Specifically, we demonstrate that, with the aid of uniform dithering, both constrained and unconstrained Lassos can stably recover signal and corruption from the quantized samples when the measurement matrix is sub-Gaussian. Our theoretical results reveal the role of quantization resolution in the recovery performance of Lassos. Numerical experiments are provided to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

This paper studies the problem of accurately recovering a structured signal from a small number of corrupted sub-Gaussian measurements. We consider three different procedures to reconstruct signal and corruption when different kinds of prior knowledge are available. In each case, we provide conditions (in terms of the number of measurements) for stable signal recovery from structured corruption with added unstructured noise. Our results theoretically demonstrate how to choose the regularization parameters in both partially and fully penalized recovery procedures and shed some light on the relationships among the three procedures. The key ingredient in our analysis is an extended matrix deviation inequality for isotropic sub-Gaussian matrices, which implies a tight lower bound for the restricted singular value of the extended sensing matrix. Numerical experiments are presented to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this letter, we characterize a data-time tradeoff for projected gradient descent (PGD) algorithms used for solving corrupted sensing problems under sub-Gaussian measurements. We also show that with a proper step size, the PGD method can achieve a linear rate of convergence when the number of measurements is sufficiently large. [ABSTRACT FROM AUTHOR]

This paper considers the problem of recovering a structured signal from a relatively small number of noisy measurements with the aid of a similar signal which is known beforehand. We propose a new approach to integrate prior information into the standard recovery procedure by maximizing the correlation between the prior knowledge and the desired signal. We then establish performance guarantees (in terms of the number of measurements) for the proposed method under sub-Gaussian measurements. Specific structured signals including sparse vectors, block-sparse vectors, and low-rank matrices are also analyzed. Furthermore, we present an interesting geometrical interpretation for the proposed procedure. Our results demonstrate that if prior information is good enough, then the proposed approach can (remarkably) outperform the standard recovery procedure. Simulations are provided to verify our results. [ABSTRACT FROM AUTHOR]

To improve the reconstruction performance of the generalized orthogonal matching pursuit, an improved method is proposed. Columns are selected from the sensing matrix by generalized orthogonal matching pursuit, and indices of the columns are added to the estimated support set to reconstruct a sparse signal. Those columns contain error columns that can reduce the reconstruction performance. Therefore, the proposed algorithm adds a backtracking process to remove the low-reliability columns from the selected column set. For any k-sparse signal, the proposed method firstly computes the correlation between the columns of the sensing matrix and the residual vector and then selects s columns that correspond to the s largest correlation in magnitude and adds their indices to the estimated support set in each iteration. Secondly, the proposed algorithm projects the measurements onto the space that consists of those selected columns and calculates the projection coefficient vector. When the size of the support set is larger than k, the proposed method will select k high-reliability indices using a search strategy from the support set. Finally, the proposed method updates the estimated support set using the selected k high-reliability indices. The simulation results demonstrate that the proposed algorithm has a better recovery performance. [ABSTRACT FROM AUTHOR]

Abstract: We provide another framework of iterative algorithms based on thresholding, feedback and null space tuning for sparse signal recovery arising in sparse representations and compressed sensing. Several thresholding algorithms with various feedbacks are derived. Convergence results are also provided. The core algorithm is shown to converge in finitely many steps under a (preconditioned) restricted isometry condition. The algorithms are seen as exceedingly effective and fast, particularly for large scale problems. Numerical studies about the effectiveness and the speed of the algorithms are also presented. [Copyright &y& Elsevier]

Compressed sensing with sparse frame representations is seen to have much greater range of practical applications than that with orthonormal bases. In such settings, one approach to recover the signal is known as \ell 1-analysis. We expand in this paper the performance analysis of this approach by providing a weaker recovery condition than existing results in the literature. Our analysis is also broadly based on general frames and alternative dual frames (as analysis operators). As one application to such a general-dual-based approach and performance analysis, an optimal-dual-based technique is proposed to demonstrate the effectiveness of using alternative dual frames as analysis operators. An iterative algorithm is outlined for solving the optimal-dual-based \ell 1-analysis problem. The effectiveness of the proposed method and algorithm is demonstrated through several experiments. [ABSTRACT FROM PUBLISHER]

The stochastic gradient matching pursuit algorithm requires the sparsity of the signal as prior information. However, this prior information is unknown in practical applications, which restricts the practical applications of the algorithm to some extent. An improved method was proposed to overcome this problem. First, a pre-evaluation strategy was used to evaluate the sparsity of the signal and the estimated sparsity was used as the initial sparsity. Second, if the number of columns of the candidate atomic matrix was smaller than that of the rows, the least square solution of the signal was calculated, otherwise, the least square solution of the signal was set as zero. Finally, if the current residual was greater than the previous residual, the estimated sparsity was adjusted by the fixed step-size and stage index, otherwise we did not need to adjust the estimated sparsity. The simulation results showed that the proposed method was better than other methods in terms of the aspect of reconstruction percentage in the larger sparsity environment. [ABSTRACT FROM AUTHOR]