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 ${\cal M}$ 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 ${\cal M}$. 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.