Improved sparse recovery thresholds with two-step reweighted ℓ1 minimization

It is well known that ℓ₁ minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity, as a function of the ratio between the system dimensions, so that with high probability almost all sparse signals can be recovered from iid Gaussian measurements, have been computed and are referred to as “weak thresholds”. In this paper, we introduce a reweighted ℓ₁ recovery algorithm composed of two steps: a standard ℓ₁ minimization step to identify a set of entries where the signal is likely to reside, and a weighted ℓ₁ minimization step where entries outside this set are penalized. For signals where the non-sparse component has iid Gaussian entries, we prove a “strict” improvement in the weak recovery threshold. Simulations suggest that the improvement can be quite impressive-over 20% in the example we consider.