On Partial Smoothness, Activity Identification and Faster Algorithms of L1 Over L2 Minimization

The L-1 /L-2 norm ratio arose as a sparseness measure and attracted a considerable amount of attention due tothree merits: (i) sharper approximations ofL0compared to theL(1); (ii) parameter-free and scale-invariant; (iii) more attractive than L-1 under highly-coherent matrices. In this paper, we firstestablish the partly smooth property of L-1 ove rL(2) minimizationrelative to an active manifold M and also demonstrate itsprox-regularity property. Second, we reveal that AD M Mp(orADMM+p) can identify the active manifold within a finite itera-tions. 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 ADMMp(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 show case the superiority of our heuristic algorithm over existing state-of-the-art methods for sparse recovery.