Linear Convergence of Adaptively Iterative Thresholding Algorithms for Compressed Sensing

TL;DR

This paper proves that the adaptively iterative thresholding (AIT) algorithm converges linearly for compressed sensing under a generalized restricted isometry property, with theoretical guarantees and simulation validation.

Contribution

It introduces a generalized RIP (gRIP) condition for analyzing AIT convergence and provides new theoretical results improving upon existing conditions.

Findings

AIT converges linearly in noise-free case under gRIP

Convergence remains linear in noisy case until a certain error bound

Simulations confirm theoretical convergence and effectiveness

Abstract

This paper studies the convergence of the adaptively iterative thresholding (AIT) algorithm for compressed sensing. We first introduce a generalized restricted isometry property (gRIP). Then we prove that the AIT algorithm converges to the original sparse solution at a linear rate under a certain gRIP condition in the noise free case. While in the noisy case, its convergence rate is also linear until attaining a certain error bound. Moreover, as by-products, we also provide some sufficient conditions for the convergence of the AIT algorithm based on the two well-known properties, i.e., the coherence property and the restricted isometry property (RIP), respectively. It should be pointed out that such two properties are special cases of gRIP. The solid improvements on the theoretical results are demonstrated and compared with the known results. Finally, we provide a series of simulations to verify the correctness of the theoretical assertions as well as the effectiveness of the AIT algorithm.