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It is now well understood that $\ell_1$ minimization algorithm is able torecover sparse signals from incomplete measurements [2], [1], [3] and sharprecoverable sparsity thresholds have also been obtained for the $\ell_1$minimization algorithm. However, even though iterative reweighted $\ell_1$minimization algorithms or related algorithms have been empirically observed toboost the recoverable sparsity thresholds for certain types of signals, norigorous theoretical results have been established to prove this fact. In thispaper, we try to provide a theoretical foundation for analyzing the iterativereweighted $\ell_1$ algorithms. In particular, we show that for a nontrivialclass of signals, the iterative reweighted $\ell_1$ minimization can indeeddeliver recoverable sparsity thresholds larger than that given in [1], [3]. Ourresults are based on a high-dimensional geometrical analysis (Grassmann angleanalysis) of the null-space characterization for $\ell_1$ minimization andweighted $\ell_1$ minimization algorithms.

Breaking through the Thresholds: an Analysis for Iterative Reweighted $\ell_1$ Minimization via the Grassmann Angle Framework