Newton-Type Optimal Thresholding Algorithms for Sparse Optimization Problems

TL;DR

This paper introduces Newton-type optimal thresholding algorithms that combine Newton methods with optimal $k$-thresholding for improved sparse signal recovery, demonstrating their stability and efficiency through theoretical analysis and simulations.

Contribution

It proposes a novel Newton-type optimal $k$-thresholding algorithm for sparse optimization, integrating Newton methods with optimal thresholding for better performance.

Findings

Algorithms are proven to converge under RIP conditions.

Simulation results show stability and efficiency in signal recovery.

Performance depends on proper parameter choices.

Abstract

Sparse signals can be possibly reconstructed by an algorithm which merges a traditional nonlinear optimization method and a certain thresholding technique. Different from existing thresholding methods, a novel thresholding technique referred to as the optimal $k$-thresholding was recently proposed by Zhao [SIAM J Optim, 30(1), pp. 31-55, 2020]. This technique simultaneously performs the minimization of an error metric for the problem and thresholding of the iterates generated by the classic gradient method. In this paper, we propose the so-called Newton-type optimal $k$-thresholding (NTOT) algorithm which is motivated by the appreciable performance of both Newton-type methods and the optimal $k$-thresholding technique for signal recovery. The guaranteed performance (including convergence) of the proposed algorithms are shown in terms of suitable choices of the algorithmic parameters and the restricted isometry property (RIP) of the sensing matrix which has been widely used in the analysis of compressive sensing algorithms. The simulation results based on synthetic signals indicate that the proposed algorithms are stable and efficient for signal recovery.