Sparse signal recovery via non-convex optimization and overcomplete dictionaries

Abstract

In this paper, we address the problem of recovering 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 relies on [Formula: see text] minimization with [Formula: see text] is exact. In addition, under a mild assumption about the dictionary [Formula: see text], we use a similar method [H. Rauhut et al., Compressed sensing and redundant dictionaries, IEEE Trans. Inf. Theory 54(5) (2008) 2210–2219] to derive an estimation of the restricted isometry constant of the composed matrix [Formula: see text]. Finally, the performance of the [Formula: see text] minimization is testified by some numerical examples.