  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.
