Unified Analysis on L1 over L2 Minimization for signal recovery

Abstract: In this paper, we carry out a unified study for $L_1$ over $L_2$ sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signals. First, we provide a unified theoretical analysis on the existence of the global solutions of the constrained and the unconstrained $L_{1}/L_{2}$ models. Second, we analyze the sparse property of any local minimizer of these $L_{1}/L_{2}$ models which serves as a certificate to rule out the nonlocal-minimizer stationary solutions. Third, we derive an analytical solution for the proximal operator of the $L_{1} / L_{2}$ with nonnegative constraint. Equipped with this, we apply the alternating direction method of multipliers to the unconstrained model with nonnegative constraint in a particular splitting way, referred to as ADMM$_p^+$. We establish its global convergence to a d-stationary solution (sharpest stationary) without the Kurdyka-\L ojasiewicz assumption. Extensive numerical simulations confirm the superior of ADMM$_p^+$ over the state-of-the-art methods in sparse recovery. In particular, ADMM$_p^+$ reduces computational time by about $95\%\sim99\%$ while achieving a much higher accuracy than the commonly used scaled gradient projection method for the wavelength misalignment problem.