Support driven reweighted $\ell_1$ minimization

In this paper, we propose a support driven reweighted $\ell_1$ minimization algorithm (SDRL1) that solves a sequence of weighted $\ell_1$ problems and relies on the support estimate accuracy. Our SDRL1 algorithm is related to the IRL1 algorithm proposed by Cand{\`e}s, Wakin, and Boyd. We demonstrate that it is sufficient to find support estimates with \emph{good} accuracy and apply constant weights instead of using the inverse coefficient magnitudes to achieve gains similar to those of IRL1. We then prove that given a support estimate with sufficient accuracy, if the signal decays according to a specific rate, the solution to the weighted $\ell_1$ minimization problem results in a support estimate with higher accuracy than the initial estimate. We also show that under certain conditions, it is possible to achieve higher estimate accuracy when the intersection of support estimates is considered. We demonstrate the performance of SDRL1 through numerical simulations and compare it with that of IRL1 and standard $\ell_1$ minimization.