Breaking the ℓ1 recovery thresholds with reweighted ℓ1 optimization

It is now well understood that l₁ minimization algorithm is able to recover sparse signals from incomplete measurements and sharp recoverable sparsity thresholds have also been obtained for the l₁ minimization algorithm. In this paper, we investigate a new iterative reweighted l₁ minimization algorithm and showed that the new algorithm can increase the sparsity recovery threshold of l₁ minimization when decoding signals from relevant distributions. Interestingly, we observed that the recovery threshold performance of the new algorithm depends on the behavior, more specifically the derivatives, of the signal amplitude probability distribution at the origin.