Sparse Signal Reconstruction for Nonlinear Models via Piecewise Rational Optimization

TL;DR

This paper introduces a global optimization method for reconstructing sparse signals affected by nonlinear distortions, using polynomial relaxation techniques to address nonconvex challenges and improve reconstruction accuracy.

Contribution

It presents a novel approach employing Lasserre relaxations for global solutions in sparse signal reconstruction with nonlinear models, including piecewise rational functions.

Findings

Achieves global optimality in sparse signal reconstruction.

Effectively handles nonlinear distortions with polynomial relaxations.

Demonstrates improved reconstruction accuracy through numerical simulations.

Abstract

We propose a method to reconstruct sparse signals degraded by a nonlinear distortion and acquired at a limited sampling rate. Our method formulates the reconstruction problem as a nonconvex minimization of the sum of a data fitting term and a penalization term. In contrast with most previous works which settle for approximated local solutions, we seek for a global solution to the obtained challenging nonconvex problem. Our global approach relies on the so-called Lasserre relaxation of polynomial optimization. We here specifically include in our approach the case of piecewise rational functions, which makes it possible to address a wide class of nonconvex exact and continuous relaxations of the $\ell_0$ penalization function. Additionally, we study the complexity of the optimization problem. It is shown how to use the structure of the problem to lighten the computational burden efficiently. Finally, numerical simulations illustrate the benefits of our method in terms of both global optimality and signal reconstruction.