Greedy Gauss-Newton algorithm for finding sparse solutions to nonlinear underdetermined systems of equations

TL;DR

This paper introduces a greedy Gauss-Newton algorithm for efficiently finding sparse solutions to underdetermined nonlinear systems, outperforming traditional  methods in tests.

Contribution

It proposes a novel greedy approach for selecting Jacobian columns in Gauss-Newton, improving convergence and efficiency for sparse solutions.

Findings

Outperforms  methods on test problems

Proven convergence and efficiency

Effective column selection strategy

Abstract

We consider the problem of finding sparse solutions to a system of underdetermined nonlinear system of equations. The methods are based on a Gauss-Newton approach with line search where the search direction is found by solving a linearized problem using only a subset of the columns in the Jacobian. The choice of columns in the Jacobian is made through a greedy approach looking at either maximum descent or an approach corresponding to orthogonal matching for linear problems. The methods are shown to be convergent and efficient and outperform the $\ell_1$ approach on the test problems presented.