Estimation of sparse signal by non-convex optimization

It is standard in compressed sensing scenarios to assume that the signal f can be sparsely represented in an orthonormal basis. Whereas, in some sense this isn't very realistic. Indeed, allowing the signal to be sparse with respect to a redundant dictionary adds a lot of flexibility and significantly extends the range of applicability. In this paper, we address the problem of recover signals from undersampled data where such signals are not sparse in an orthonormal basis, but in an overcomplete dictionary. We show that if the combined matrix obeys a certain restricted isometry property and if the signal is sufficiently sparse, the reconstruction that rely on ℓ_p minimization with 0 < p < 1 is exact.