On Partly Smoothness, Activity Identification and Faster Algorithms of L_1 over L_2 Minimization

The L_1/L_2 norm ratio arose as a sparseness measure and attracted a considerable amount of attention due to three merits: (i) sharper approximations of L_0 compared to the L_1; (ii) parameter-free and scale-invariant; (iii) more attractive than L_1 under highly-coherent matrices. In this paper, we first establish the partly smooth property of L_1 over L_2 minimization relative to an active manifold M and also demonstrate its prox-regularity property. Second, we reveal that ADMM_p (or ADMM^+_p) can identify the active manifold within a finite iterations. This discovery contributes to a deeper understanding of the optimization landscape associated with L_1 over L_2 minimization. Third, we propose a novel heuristic algorithm framework that combines ADMM_p (or ADMM^+_p) with a globalized semismooth Newton method tailored for the active manifold M. This hybrid approach leverages the strengths of both methods to enhance convergence. Finally, through extensive numerical simulations, we showcase the superiority of our heuristic algorithm over existing state-of-the-art methods for sparse recovery.