Weighted one-norm minimization with inaccurate support estimates: Sharp analysis via the null-space property

We study the problem of recovering sparse vectors given possibly erroneous support estimates. First, we provide necessary and sufficient conditions for weighted ℓ₁ minimization to successfully recovery all sparse signals whose support estimate is sufficiently accurate. We relate these conditions to the analogous ones for ℓ₁ minimization, showing that they are equivalent when the support estimate is 50% accurate but that the weighted ℓ₁ conditions are easier to satisfy when the support is more than 50% accurate. Second, to quantify this improvement, we provide bounds on the number of Gaussian measurements that ensure, with high probability, that weighted ℓ₁ minimization succeeds. The resulting number of measurements can be significantly less than what is needed to ensure recovery via ℓ₁ minimization. Finally, we illustrate our results via numerical experiments.