Sparse signal recovery via minimax-concave penalty and ℓ1-norm loss function

In sparse signal recovery, to overcome the ℓ₁-norm sparse regularisation's disadvantages tendency of uniformly penalise the signal amplitude and underestimate the high-amplitude components, a new algorithm based on a non-convex minimax-concave penalty is proposed, which can approximate the ℓ₀-norm more accurately. Moreover, the authors employ the ℓ₁-norm loss function instead of the ℓ₂-norm for the residual error, as the ℓ₁-loss is less sensitive to the outliers in the measurements. To rise to the challenges introduced by the non-convex non-smooth problem, they first employ a smoothed strategy to approximate the ℓ₁-norm loss function, and then use the difference-of-convex algorithm framework to solve the nonconvex problem. They also show that any cluster point of the sequence generated by the proposed algorithm converges to a stationary point. The simulation result demonstrates the authors' conclusions and indicates that the algorithm proposed in this study can obviously improve the reconstruction quality.