A linearly convergent algorithm for sparse signal reconstruction

Dexiang Feng,Xueyong Wang

doi:10.1007/s11784-018-0635-1

Abstract

For the sparse signal reconstruction problem in compressive sensing, we propose a projection-type algorithm without any backtracking line search based on a new formulation of the problem. Under suitable conditions, global convergence and its linear convergence of the designed algorithm are established. The efficiency of the algorithm is illustrated through some numerical experiments on some sparse signal reconstruction problem.

Highlights

A basic mathematical problem in compressive sensing (CS) is to recover a sparse signal vector x ∈ Rn from an undetermined linear system y = Ax, where A ∈ Rm×n (m n) is the sensing matrix
For the sparse signal reconstruction problem in compressive sensing, we propose a projection-type algorithm without any backtracking line search based on a new formulation of the problem
The efficiency of the algorithm is illustrated through some numerical experiments on some sparse signal reconstruction problem

Summary

Introduction

For convex optimization problem (1.1), there are some standard methods such as smooth Newton-type methods and interior-point methods for solving the Yin et al [66] proposed an efficient method for solving the 1-minimization problem based on Bregman iterative regularization. Hale et al [16] presented a framework for solving the large-scale 1-regularized convex minimization problem based on operator splitting and continuation. These solvers are not tailored for large-scale cases of CS and they become inefficient as dimension n increases. We use R+n to denote the nonnegative quadrant in Rn, and use x+ to denote the orthogonal projection of vector x ∈ Rn onto R+n , that is, (x+)i := max{xi, 0}, 1 ≤ i ≤ n; the norm · and · 1 denote the Euclidean 2-norm and 1-norm, respectively

New formulation and algorithm

Convergence

Numerical experiments

Conclusion