A differential equations approach to $l_1$-minimization with applications to array imaging

TL;DR

This paper introduces an ODE-based framework for analyzing $l_1$-minimization algorithms, enabling convergence proofs and demonstrating effectiveness in sparse array imaging applications.

Contribution

It develops a novel ODE approach for $l_1$-minimization analysis, connecting continuous dynamics with discrete algorithms for improved understanding and performance.

Findings

Energy methods prove convergence of the ODE approach.

The discrete algorithm performs effectively in sparse array imaging.

The relaxation parameter does not affect the minimum of the relaxed problem.

Abstract

We present an ordinary differential equations approach to the analysis of algorithms for constructing $l_1$ minimizing solutions to underdetermined linear systems of full rank. It involves a relaxed minimization problem whose minimum is independent of the relaxation parameter. An advantage of using the ordinary differential equations is that energy methods can be used to prove convergence. The connection to the discrete algorithms is provided by the Crandall-Liggett theory of monotone nonlinear semigroups. We illustrate the effectiveness of the discrete optimization algorithm in some sparse array imaging problems.