The variant of the iterative shrinkage-thresholding algorithm for minimization of the ℓ1 over ℓ∞ norms

Abstract

In this paper, we study minimization of the ratio of ℓ1 and ℓ∞ norms (ℓ1/ℓ∞) as a nonconvex and sparsity-promoting metric for solving the unconstrained compressed sensing problems. To design some efficient algorithms, we derive a closed-form solution to the proximal operator of the ℓ1/ℓ∞ function. With the newly variant of the shrinkage operator, we propose a novel variant of fast iterative shrinkage-thresholding algorithm (FISTA) and a specific variable-splitting scheme of the alternating direction method of multipliers (ADMM) which is guaranteed to have sub-sequential convergence. Experimentally, we construct extensive numerical simulations to demonstrate the efficiency of these two proposed approaches over the state-of-the-art algorithms in sparse recovery.