Enhancing convergence speed in sparse signal denoising: the APFISTA algorithm

Abstract

Among numerous first-order algorithms, the Fast Iterative Shrinkage-Thresholding Algorithm, known as FISTA, is renowned for its convergence speed of O ( 1 k 2 ) in terms of the objective function value, where k denotes the number of iterations. Additionally, various improvements to FISTA have been proposed in the literature. Among them, the convergence rate of a parameterized FISTA (PFISTA) proposed by Liang, Luo and Schonlieb [SIAM J Sci Comput. 2022:44(3):A1069–A1091] is obviously faster than that of the original FISTA, which can reach o ( 1 k 2 ) convergence rate. Building upon the idea of parameterization, we introduce the fully parameterized APFISTA for complete parameterization of the inertia term, aiming to enhance the current situation of partially parameterized inertia terms. We establish the convergence speed of its objective function value and the sequence generated by the algorithm. Finally, we apply the proposed method to solve the least squares problem and linear inverse problems. The obtained numerical results illustrate the practical behavior and theoretical analysis of the proposed approach.