Breaking through the thresholds: an analysis for iterative reweighted ℓ1 minimization via the Grassmann angle framework

It is now well understood that the 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. However, even though iterative reweighted ℓ₁ minimization algorithms or related algorithms have been empirically observed to boost the recoverable sparsity thresholds for certain types of signals, no rigorous theoretical results have been established to prove this fact. In this paper, we try to provide a theoretical foundation for analyzing the iterative reweighted ℓ₁ algorithms. In particular, we show that for a nontrivial class of signals, the iterative reweighted ℓ₁ minimization can indeed deliver recoverable sparsity thresholds larger than that given in. Our results are based on a high-dimensional geometrical analysis (Grassmann angle analysis) of the null-space characterization for ℓ₁ minimization and weighted ℓ₁ minimization algorithms.