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

Unified Analysis on L1 over L2 Minimization for signal recovery