Sparse Recovery with Partial Support Knowledge

The goal of sparse recovery is to recover the (approximately) best k-sparse approximation x̂ of an n-dimensional vector x from linear measurements Ax of x. We consider a variant of the problem which takes into account partial knowledge about the signal. In particular, we focus on the scenario where, after the measurements are taken, we are given a set S of size s that is supposed to contain most of the “large” coefficients of x. The goal is then to find x̂ such that ‖x− x̂‖p ≤ C min k-sparse x′ supp(x′)⊆S ‖x− x‖q . (1) We refer to this formulation as the sparse recovery with partial support knowledge problem (SRPSK). We show that SRPSK can be solved, up to an approximation factor of C = 1 + , using O((k/ ) log(s/k)) measurements, for p = q = 2. Moreover, this bound is tight as long as s = O( n/ log(n/ )). This completely resolves the asymptotic measurement complexity of the problem except for a very small range of the parameter s. To the best of our knowledge, this is the first variant of (1+ )-approximate sparse recovery for which the asymptotic measurement complexity has been determined.