Stable reconstructions for the analysis formulation of ℓp-minimization using redundant systems

A common method to compute sparse solutions in compressed sensing is based on `minimization. In this paper we consider `-minimization for arbitrary 0 < p ≤ 1. More precisely, we prove stability estimates for solutions of the analysis formulation of the `minimization problem for 0 < p ≤ 1 from noisy measurements. Furthermore, our focus lies in arbitrary frames that are not necessarily tight or Parseval. We will investigate the role of the frame bounds and how they effect the sparsity as well as the number of measurements that are needed. Furthermore, we will distinguish between the minimization over frame coefficients and dual frame coefficients. Numerical experiments based on iterative reweighting are also presented that suggest the possible benefit of using `-minimization over `-minimization for sparse recovery for some applications.