Support Recovery From Noisy Random Measurements via Weighted $\ell _1$ Minimization

Herein, the sample complexity of general weighted $\ell _1$ minimization in terms of support recovery from noisy underdetermined measurements is analyzed. This analysis generalizes prior work on $\ell _1$ minimization by considering arbitrary weighting. The explicit relationship between the weights and the sample complexity is stated such that for random matrices with i.i.d. Gaussian entries, the weighted $\ell _1$ 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 reweighted strategy is proposed for binary signal recovery. In the binary sparsity-Promoting Reweighted $\ell _1$ minimization (bPRL1) algorithm, a sequence of weighted $\ell _1$ 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. bPRL1 is compared to other weighted algorithms through the proposed measure and numerical results are shown to provide superior performance for a spectrum occupancy estimation problem motivated by cognitive radio.