A Scale Invariant Approach for Sparse Signal Recovery
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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 metric. Compared to popular models in the literature such as the L_p model for p∈(0,1) and the transformed L_1 (TL1), this ratio model is parameter free. Theoretically, we present a weak null space property (wNSP) and prove that any sparse vector is a local minimizer of the L_1 /L_2 model provided with this wNSP 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 MRI reconstruction.construction.
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