On Partly Smoothness, Activity Identification and Faster Algorithms of L1 over L2 Minimization

The L1/L2 norm ratio arose as a sparseness measure and attracted a considerable amount of attention due to three merits: (i) sharper approximations of L0 compared to the L1; (ii) parameter-free and scale-invariant; (iii) more attractive than L1 under highly-coherent matrices. In this paper, we first establish the partly smooth property of L1 over L2 minimization relative to an active manifold cal M and also demonstrate its prox-regularity property. Second, we reveal that ADMMp (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 L1 over L2 minimization. Third, we propose a novel heuristic algorithm framework that combines ADMMp (or ADMM^+p) with a globalized semismooth Newton method tailored for the active manifold cal 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.