A Scale Invariant Approach for Sparse Signal Recovery
In this paper, we study the ratio of the $L_1 and $L_2 norms, denoted as $L_1/L_2$ 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 $L_p$ model for $p\in(0,1) and the transformed $L_1$ (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 $L_1 /L_2 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 $L_1/L_2$ model to apply on the gradient is also discussed with a proof of concept example of the MRI reconstruction.
SIAM J. Sci. Comput.