A Convergent Fixed-Point Proximity Algorithm Accelerated by FISTA for the ℓ₀ Sparse Recovery Problem

We propose an approximation model of the original ℓ₀ minimization model arising from various sparse signal recovery problems. The objective function of the proposed model uses the Moreau envelope of the ℓ₀ norm to promote the sparsity of the signal in a tight framelet system. This leads to a non-convex optimization problem involved the ℓ₀ norm. We identify a local minimizer of the proposed non-convex optimization problem with a global minimizer of a related convex optimization problem. Based on this identification, we develop a two stage algorithm for solving the proposed non-convex optimization problem and study its convergence. Moreover, we show that FISTA can be employed to speed up the convergence rate of the proposed algorithm to reach the optimal convergence rate of O(1 ∕ k²). We present numerical results to confirm the theoretical estimate.