On Partial 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.