An $l_{1}-l_{1}$ -Norm Minimization Solution Using ADMM with FISTA

This paper discusses compressed sensing which reconstructs original sparse signal from observed data. Our approach formulates the weighted sum of $l_{1}$ -norm error and $l_{1}$ -norm regularization terms, and applies Alternating Direction Method of Multipliers (ADMM) to solve it. Many works employ ADMM for the $l_{1}-l_{1}$ -norm minimization problems, where ADMM obtains solutions in an iterative fashion for the problems formed as an augmented Lagrangian. The ADMM process is divided into three steps: an error minimization, a coefficient-norm minimization, and a dual variable update of an augmented Lagrangian. However, the coefficient-minimization step is not clear and replaced with an approximation. Our contribution is to adopt the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) for the minimization step and achieves faster implementation than a conventional method.