Recovery threshold for optimal weight ℓ1 minimization

We consider the problem of recovering a sparse signal from underdetermined measurements when we have prior information about the sparsity structure of the signal. In particular, we assume that the entries of the signal can be partitioned into two known sets S₁, S2 where the relative sparsities over the two sets are different. In this situation it is advantageous to replace classical ℓ₁ minimization with weighted ℓ₁ minimization, where the sparser set is given a larger weight. In this paper we give a simple closed form expression for the minimum number of measurements required for successful recovery when the optimal weights are chosen. The formula shows that the number of measurements is upper bounded by the sum of the minimum number of measurements needed had we measured the S₁ and S₂ components of the signal separately. In fact, our results indicate that this upper bound is tight and we actually have equality. Our proof technique uses the “escape through a mesh” framework and connects to the Minimax MSE of a certain basis pursuit denisoing problem.