A Unified Study on $L_1$ over $L_2$ Minimization

In this paper, we carry out a unified study for L1 over L2 sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signal. First, we provide the exact recovery condition on both the constrained and the unconstrained models for a broad set of signals. Next, we prove the solution existence of these L1/L2 models under the assumption that the null space of the measurement matrix satisfies the s-spherical section property. Then by deriving an analytical solution for the proximal operator of the L1/L2 with nonnegative constraint, we develop a new alternating direction method of multipliers based method (ADMM p ) to solve the unconstrained model. We establish its global convergence to a d-stationary solution (sharpest stationary) and its local linear convergence under certain conditions. Numerical simulations on two specific applications confirm the superior of ADMM p over the state-of-the-art methods in sparse recovery. ADMM p reduces computational time by about 95% ∼ 99% while achieving a much higher accuracy compared to commonly used scaled gradient projection method for wavelength misalignment problem. Index Terms Sparse recovery, the ratio of L1 over L2, alternating direction method of multipliers, coherent dictionary, dstationarity