Accelerated Schemes for the L1/L2 Minimization

@IEEE Transactions on Signal Processing

In this paper, we consider the L1/L2 minimization for sparse recovery and study its relationship with the L1-αL2 model. Based on this relationship, we propose three numerical algorithms to minimize this ratio model, two of which work as adaptive schemes and greatly reduce the computation time. Focusing on the two adaptive schemes, we discuss their connection to existing approaches and analyze their convergence. The experimental results demonstrate that the proposed algorithms are comparable to state-of-the-art methods in sparse recovery and work particularly well when the ground-truth signal has a high dynamic range. Lastly, we reveal some empirical evidence on the exact L1 recovery under various combinations of sparsity, coherence, and dynamic ranges, which calls for theoretical justification in the future.

A scale-invariant approach for sparse signal recovery

@SIAM Journal on Scientific Computing

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.