Nonlinear Basis Pursuit

TL;DR

This paper introduces a convex algorithm for nonlinear basis pursuit, extending compressive sensing to higher-order nonlinear systems and broadening its applicability beyond linear models.

Contribution

It presents the first convex, non-greedy algorithm for nonlinear basis pursuit, enabling sparse solutions in nonlinear measurement systems.

Findings

Algorithm successfully finds sparse solutions in nonlinear systems.

Extends compressive sensing to nonlinear measurement models.

Broadens application scope of sparse recovery methods.

Abstract

In compressive sensing, the basis pursuit algorithm aims to find the sparsest solution to an underdetermined linear equation system. In this paper, we generalize basis pursuit to finding the sparsest solution to higher order nonlinear systems of equations, called nonlinear basis pursuit. In contrast to the existing nonlinear compressive sensing methods, the new algorithm that solves the nonlinear basis pursuit problem is convex and not greedy. The novel algorithm enables the compressive sensing approach to be used for a broader range of applications where there are nonlinear relationships between the measurements and the unknowns.