A Unified Study on L_1 over L_2 Minimization
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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%∼99% while achieving a much higher accuracy compared to commonly used scaled gradient projection method for wavelength misalignment problem.
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