Nonmonotone Barzilai-Borwein Gradient Algorithm for $\ell_1$-Regularized Nonsmooth Minimization in Compressive Sensing

TL;DR

This paper introduces a nonmonotone Barzilai-Borwein gradient algorithm tailored for solving large-scale $ ext{l}_1$-regularized nonsmooth minimization problems, demonstrating superior performance in compressive sensing applications.

Contribution

It proposes a novel gradient algorithm with a nonmonotone line search for $ ext{l}_1$-regularized problems, ensuring global convergence and improved efficiency over existing methods.

Findings

Algorithm performs well on nonconvex problems from CUTEr library.

Outperforms several recent state-of-the-art algorithms in compressive sensing.

Demonstrates effectiveness in large-scale $ ext{l}_1$-regularized least squares problems.

Abstract

This paper is devoted to minimizing the sum of a smooth function and a nonsmooth $\ell_1$-regularized term. This problem as a special cases includes the $\ell_1$-regularized convex minimization problem in signal processing, compressive sensing, machine learning, data mining, etc. However, the non-differentiability of the $\ell_1$-norm causes more challenging especially in large problems encountered in many practical applications. This paper proposes, analyzes, and tests a Barzilai-Borwein gradient algorithm. At each iteration, the generated search direction enjoys descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the $\ell_1$-norm. Moreover, a nonmonotone line search technique is incorporated to find a suitable stepsize along this direction. The algorithm is easily performed, where the values of the objective function and the gradient of the smooth term are required at per-iteration. Under some conditions, the proposed algorithm is shown to be globally convergent. The limited experiments by using some nonconvex unconstrained problems from CUTEr library with additive $\ell_1$-regularization illustrate that the proposed algorithm performs quite well. Extensive experiments for $\ell_1$-regularized least squares problems in compressive sensing verify that our algorithm compares favorably with several state-of-the-art algorithms which are specifically designed in recent years.