Surpassing the theoretical 1-norm phase transition in compressive sensing by tuning the smoothed l0 algorithm

Reconstruction of an undersampled signal is at the root of compressive sensing: when is an algorithm capable of reconstructing the signal? what quality is achievable? and how much time does reconstruction require? We have considered the worst-case performance of the smoothed ℓ₀ norm reconstruction algorithm in a noiseless setup. Through an empirical tuning of its parameters, we have improved the phase transition (capabilities) of the algorithm for fixed quality and required time. In this paper, we present simulation results that show a phase transition surpassing that of the theoretical ℓ₁ approach: the proposed modified algorithm obtains 1-norm phase transition with greatly reduced required computation time.