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

TL;DR

This paper provides a rigorous convergence proof for the Zero-point Attracting Projection (ZAP) algorithm used in sparse signal recovery, analyzing its performance, robustness, and behavior in noisy conditions.

Contribution

It offers the first theoretical convergence proof for ZAP, establishes conditions for convergence, and analyzes its performance in noisy environments.

Findings

ZAP converges under specified conditions.

ZAP is non-biased and approaches the sparse solution with proper step-size.

Measurement noise linearly affects recovery precision.

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.