Sparse Signal Reconstruction for Nonlinear Models via Piecewise Rational Optimization
This addresses signal processing challenges in applications like communications or imaging, though it appears incremental as an extension of existing global optimization methods to piecewise rational functions.
The paper tackles sparse signal reconstruction from nonlinear distortions at limited sampling rates by formulating it as a nonconvex minimization problem and seeking global solutions via Lasserre relaxation, showing benefits in global optimality and reconstruction.
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.