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

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 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 $L_{1}/L_{2}$ 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 $L_{1} / L_{2}$ 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 $95\%\sim99\%$ while achieving a much higher accuracy compared to commonly used scaled gradient projection method for wavelength misalignment problem.