Proof of Convergence and Performance Analysis for Sparse Recovery via Zero-point Attracting Projection

Abstract

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.

29 pages, 6 figures

References in corpus (3)

• Sparsity and Incoherence in Compressive Sampling
• A stochastic gradient approach on compressive sensing signal reconstruction based on adaptive filtering framework
• Compressed Sensing: How sharp is the Restricted Isometry Property

Cited by in corpus (6)

• On the Performance Bound of Sparse Estimation with Sensing Matrix Perturbation
• Robustness of Sparse Recovery via $F$-minimization: A Topological Viewpoint
• Oracle-order Recovery Performance of Greedy Pursuits with Replacement against General Perturbations
• An Improved Variable Step-size Zero-point Attracting Projection Algorithm
• A New Variable Step-size Zero-point Attracting Projection Algorithm
• The Convergence Guarantees of a Non-convex Approach for Sparse Recovery