Support recovery from noisy random measurements via weighted ℓ1 minimization

Herein, we analyze the sample complexity of general weighted ℓ
₁
minimization in terms of support recovery from noisy underdetermined measurements. This analysis generalizes prior work for standard ℓ
₁
minimization by considering the weighting effect. We state explicit relationship between the weights and the sample complexity such that i.i.d random Gaussian measurement matrices used with weighted ℓ
₁
minimization recovers the support of the underlying signal with high probability as the problem dimension increases. This result provides a measure that is predictive of relative performance of different algorithms. Motivated by the analysis, a new iterative weighted strategy is proposed. In the Reweighted Partial Support (RePS) algorithm, a sequence of weighted ℓ
₁
minimization problems are solved where partial support recovery is used to prune the optimization; furthermore, the weights used for the next iteration are updated by the current estimate. RePS is compared to other weighted algorithms through the proposed measure and numerical results, which demonstrate its superior performance for a spectrum occupancy estimation problem motivated by cognitive radio.