Adaptive step-size fast iterative shrinkage-thresholding algorithm and sparse-spike deconvolution

The standard Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) adopts a search approach consisting of linear increases to determine the step size of the internal gradient. If the input of the initial step-size is not accurate, the convergence of FISTA may be restricted when the linear search scheme is applied. To overcome this problem, we tentatively reduce the step size before each iteration to then obtain the most suitable step-size using a linear search approach. To ensure the convergence of the algorithm, we introduce the step size for the previous and subsequent iterations during the calculation process. This has allowed us sparse-spike deconvolution based on an adaptive step size algorithm (ASFISTA), which to a certain extent solves the problem of the degree of convergence of the standard method. In this paper we first present the new algorithm and then we test its convergence. In order to check the effectiveness of the modified algorithm, we use both the standard FISTA method and the improved ASFISTA method to conduct sparse spike deconvolution on a theoretical model. Finally, we carry out a similar analysis aimed at the recovery of the sparse real signal.