( 1 + ) - approximate Sparse Recovery

The problem central to sparse recovery and compressive sensing is that of stable sparse recovery: we want a distribution A of matrices A ∈ Rm×n such that, for any x ∈ R and with probability 1−δ > 2/3 over A ∈ A, there is an algorithm to recover x̂ from Ax with ‖x̂− x‖p ≤ C min k-sparse x′ ∥∥x− x′∥∥ p (1) for some constant C > 1 and norm p. The measurement complexity of this problem is well understood for constant C > 1. However, in a variety of applications it is important to obtain C = 1+ for a small > 0, and this complexity is not well understood. We resolve the dependence on in the number of measurements required of a k-sparse recovery algorithm, up to polylogarithmic factors for the central cases of p = 1 and p = 2. Namely, we give new algorithms and lower bounds that show the number of measurements required is k/ polylog(n). For p = 2, our bound of 1 k log(n/k) is tight up to constant factors. We also give matching bounds when the output is required to be k-sparse, in which case we achieve k/ polylog(n). This shows the distinction between the complexity of sparse and nonsparse outputs is fundamental.