Projected ℓ1-minimization for compressed sensing

We propose a new algorithm to recover a sparse signal from a system of linear measurements. By projecting the measured signal onto a properly chosen subspace, we can use the projection to zero in on a low-sparsity portion of our original signal, which we can recover using ℓ₁-minimization. We can then recover the remaining portion of our signal from an overdetermined system of linear equations. We prove that our scheme improves the threshold of ℓ₁-minimization, and we derive an upper bound for this new threshold. We support our theoretical results with numerical simulations which demonstrate that certain classes of signals come close to achieving this upper bound.