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We study the recovery of sparse signals from underdetermined linearmeasurements when a potentially erroneous support estimate is available. Ourresults are twofold. First, we derive necessary and sufficient conditions forsignal recovery from compressively sampled measurements using weighted$\ell_1$-norm minimization. These conditions, which depend on the choice ofweights as well as the size and accuracy of the support estimate, are on thenull space of the measurement matrix. They can guarantee recovery even whenstandard $\ell_1$ minimization fails. Second, we derive bounds on the number ofGaussian measurements for these conditions to be satisfied, i.e., for weighted$\ell_1$ minimization to successfully recover all sparse signals whose supporthas been estimated sufficiently accurately. Our bounds show that weighted$\ell_1$ minimization requires significantly fewer measurements than standard$\ell_1$ minimization when the support estimate is relatively accurate.

Recovery Analysis for Weighted $\ell_1$-Minimization Using a Null Space Property