Convergence of the Generalized Alternating Projection Algorithm for Compressive Sensing

TL;DR

This paper proves the linear convergence of the generalized alternating projection (GAP) algorithm for compressive sensing under RIP conditions and shows it converges faster than adaptive iterative thresholding, supported by simulations.

Contribution

The paper provides the first theoretical proof of GAP's linear convergence under RIP and compares its convergence rate favorably against AIT.

Findings

GAP converges linearly under RIP with invertible AAmatrix.

GAP's convergence rate is faster than AIT under the same conditions.

Simulation results confirm the theoretical convergence and speed advantages.

Abstract

The convergence of the generalized alternating projection (GAP) algorithm is studied in this paper to solve the compressive sensing problem $\yv = \Amat \xv + \epsilonv$. By assuming that $\Amat\Amat\ts$ is invertible, we prove that GAP converges linearly within a certain range of step-size when the sensing matrix $\Amat$ satisfies restricted isometry property (RIP) condition of $\delta_{2K}$, where $K$ is the sparsity of $\xv$. The theoretical analysis is extended to the adaptively iterative thresholding (AIT) algorithms, for which the convergence rate is also derived based on $\delta_{2K}$ of the sensing matrix. We further prove that, under the same conditions, the convergence rate of GAP is faster than that of AIT. Extensive simulation results confirm the theoretical assertions.