Grouped sparse signal reconstruction using non-convex regularizers

This paper introduces the usage of non-convex based regularizers to solve the underdetermined MEG inverse problem. The signal to be reconstructed is considered to have a structure which entails group-wise sparsity and within group sparsity among its covariates. We discuss the usage of ℓ₂ norm regularization and smoothed ℓ₀ (SL0) norm regularization to impose group-wise and within group sparsity respectively. In addition, we introduce a novel criterion which if satisfied, guarantees global optimality while solving this non-convex optimization problem. We use proximal gradient descent as the method of optimization as it promises faster convergence rates. We show that our algorithm can successfully recover a sparse signal with a smaller number of measurements than the conventional ℓ₁ regularization framework.