A Barzilai-Borwein $l_1$-Regularized Least Squares Algorithm for Compressed Sensing

TL;DR

This paper introduces a novel Barzilai-Borwein $l_1$-regularized least squares algorithm tailored for compressed sensing, enhancing sparse solution recovery in under-determined linear systems with adaptive step sizing.

Contribution

It presents a new algorithm combining Barzilai-Borwein projection with $l_1$-regularization, offering improved step length selection for sparse signal reconstruction.

Findings

The algorithm effectively recovers sparse signals in compressed sensing.

Numerical experiments show competitive performance compared to existing methods.

Adaptive step length improves convergence speed and solution accuracy.

Abstract

Problems in signal processing and medical imaging often lead to calculating sparse solutions to under-determined linear systems. Methodologies for solving this problem are presented as background to the method used in this work where the problem is reformulated as an unconstrained convex optimization problem. The least squares approach is modified by an $l_1$-regularization term. A sparse solution is sought using a Barzilai-Borwein type projection algorithm with an adaptive step length. New insight into the choice of step length is provided through a study of the special structure of the underlying problem. Numerical experiments are conducted and results given, comparing this algorithm with a number of other current algorithms.