A Scale Invariant Approach for Sparse Signal Recovery
This work addresses sparse recovery in signal processing, offering a parameter-free alternative to existing models, but it is incremental as it builds on known non-convex approaches.
The paper tackles the problem of sparse signal recovery by proposing a scale-invariant model using the ratio of L1 and L2 norms, which is parameter-free and shown to have local minimizers for sparse vectors under a strong null space property, with experiments indicating it is comparable to state-of-the-art methods.
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.