New algorithms for sparse representation of discrete signals based on ℓp-ℓ2 optimization

This paper investigates a nonconvex relaxation of the popular ℓ₁-ℓ₂ formulation for finding sparse representation of discrete signals in overcomplete dictionary. The specific nonconvex problem we propose to solve is an ℓ_p-ℓ₂ problem with 0 <; p <; 1. Our algorithms are built on a recent algorithm, known as the monotone fast iterative shrinkage/thresholding algorithm, where a key step of soft shrinkage is replaced by a global solver for the minimization of a 1-D nonconvex ℓ_p problem. Two efficient techniques for solving the 1-D ℓ_p problem in question are proposed. Simulation studies are presented to evaluate the performance of the proposed algorithms with various values of p and compare with the well known basis pursuit (BP) algorithm with p = 1.