A Scale Invariant Approach for Sparse Signal Recovery

In this paper, we study the ratio of the L1 and L2 norms, denoted as L1/L2, to promote sparsity. Due to the non-convexity and non-linearity, there has been little attention to this scale-invariant model. Compared to popular models in the literature such as the Lp model for p∈(0,1) and the transformed L1 (TL1), this ratio model is parameter free. Theoretically, we present a strong null space property (sNSP) and prove that any sparse vector is a local minimizer of the L1 /L2 model provided with this sNSP condition. Computationally, we focus on a constrained formulation that can be solved via the alternating direction method of multipliers (ADMM). Experiments show that the proposed approach is comparable to the state-of-the-art methods in sparse recovery. In addition, a variant of the L1/L2 model to apply on the gradient is also discussed with a proof-of-concept example of the MRI reconstruction.