Faithful short-time acquisition of a sparse signal is still a challenging issue. Instead of an idealized sampling, one has only access to an altered version of it through a measurement system. This paper proposes a reconstruction method for the original sparse signal when the measurement degradation is composed of a nonlinearity, an additive noise, and a sub-sampling scheme. A rational criterion based on a least-squares fitting penalized with a suitable approximation of l0 is minimized using a recent approach guaranteeing global optimality for rational optimization. We provide a complexity analysis and show that the sub-sampling offers a significant gain in terms of computational time. This allows us to tackle practical problems such as chromatography. Finally, experimental results illustrate that our method compares very favorably to existing methods in terms of accuracy in the signal reconstruction.