In the area of sparse recovery, numerous researches hint that non-convex penalties might induce better sparsity than convex ones, but up until now those corresponding non-convex algorithms lack convergence guarantees from the initial solution to the global optimum. This paper aims to provide performance guarantees of a non-convex approach for sparse recovery. Specifically, the concept of weak convexity is incorporated into a class of sparsity-inducing penalties to characterize the non-convexity. Borrowing the idea of the projected subgradient method, an algorithm is proposed to solve the nonconvex optimization problem. In addition, a uniform approximate projection is adopted in the projection step to make this algorithm computationally tractable for large scale problems. The convergence analysis is provided in the noisy scenario. It is shown that if the non-convexity of the penalty is below a threshold (which is in inverse proportion to the distance between the initial solution and the sparse signal), the recovered solution has recovery error linear in both the step size and the noise term. Numerical simulations are implemented to test the performance of the proposed approach and verify the theoretical analysis.

Orthogonal Matching Pursuit (OMP) is a canonical greedy pursuit algorithm for sparse approximation. Previous studies of OMP have considered the recovery of a sparse signal x through Φ and y = Φx + b, where Φ is a matrix with more columns than rows and b denotes the measurement noise. In this paper, based on Restricted Isometry Property (RIP), the performance of OMP is analyzed under general perturbations, which means both y and Φ are perturbed. Though the exact recovery of an almost sparse signal x is no longer feasible, the main contribution reveals that the support set of the best k-term approximation of x can be recovered under reasonable conditions. The error bound between x and the estimation of OMP is also derived. By constructing an example it is also demonstrated that the sufficient conditions for support recovery of the best k-term approximation of x are rather tight. When x is strong-decaying, it is proved that the sufficient conditions for support recovery of the best k-term approximation of x can be relaxed, and the support can even be recovered in the order of the entries' magnitude. Our results are also compared in detail with some related previous ones.

As a canonical greedy algorithm, Orthogonal Matching Pursuit (OMP) is used for sparse approximation. Previous studies have mainly considered non-perturbed observations y = Φx, and focused on the exact recovery of x through y and Φ. Here, Φ is a matrix with more columns than rows, and x is a sparse signal to be recovered. This paper deals with performance of OMP under general perturbations-from both y and Φ. The main contribution shows that exact recovery of the support set of x can be guaranteed under suitable conditions. Such conditions are RIP-based, and involve the concept of sparsity, relative perturbation, and the smallest nonzero entry. In addition, certain conditions are given under which the support set of x can be reconstructed in the order of its entries' magnitude. In the end, it is pointed out that the conditions can be relaxed at the expense of a decrease in the accuracy of the recovery.

Compressive sensing (CS) is a data acquisition technique that measures sparse or compressible signals at a sampling rate lower than their Nyquist rate. Results show that sparse signals can be reconstructed using greedy algorithms, often requiring prior knowledge such as the signal sparsity or the noise level. As a substitute to prior knowledge, cross validation (CV), a statistical method that examines whether a model overfits its data, has been proposed to determine the stopping condition of greedy algorithms. This paper analyses cross validation in a general compressive sensing framework. Furthermore, we provide both theoretical analysis and numerical simulations for a cross-validation modification of orthogonal matching pursuit, referred to as OMP-CV, which has good performance in sparse recovery. ICASSP 2014This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. that sparse signals can be reconstructed using greedy algorithms, often requiring prior knowledge such as the signal sparsity or the noise level. As a substitute to prior knowledge, cross validation (CV), a statistical method that examines whether a model overfits its data, has been proposed to determine the stopping condition of greedy algorithms. This paper analyses cross validation in a general compressive sensing framework. Furthermore, we provide both theoretical analysis and numerical simulations for a cross-validation modification of orthogonal matching pursuit, referred to as OMP-CV, which has good performance in sparse recovery.

A recursive algorithm named Zero-point Attracting Projection (ZAP) is proposed recently for sparse signal reconstruction. Compared with the reference algorithms, ZAP demonstrates rather good performance in recovery precision and robustness. However, any theoretical analysis about the mentioned algorithm, even a proof on its convergence, is not available. In this work, a strict proof on the convergence of ZAP is provided and the condition of convergence is put forward. Based on the theoretical analysis, it is further proved that ZAP is non-biased and can approach the sparse solution to any extent, with the proper choice of step-size. Furthermore, the case of inaccurate measurements in noisy scenario is also discussed. It is proved that disturbance power linearly reduces the recovery precision, which is predictable but not preventable. The reconstruction deviation of p-compressible signal is also provided. Finally, numerical simulations are performed to verify the theoretical analysis.