A Gradient-thresholding Algorithm for Sparse Regularization

TL;DR

This paper introduces a new gradient-thresholding iterative regularization algorithm that improves accuracy and sparsity in inverse problems, with a practical stopping criterion and demonstrated efficiency in image deblurring tasks.

Contribution

A novel gradient-thresholding regularization method that is simpler and more effective than existing algorithms like FISTA, with a practical stopping rule for sparse inverse problem solutions.

Findings

Achieves better accuracy and sparsity in solutions

Outperforms standard methods like TV, FISTA, LSQR in experiments

Demonstrates computational efficiency in image deblurring

Abstract

Inverse problems arise in a wide spectrum of applications in fields ranging from engineering to scientific computation. Connected with the rise of interest in inverse problems is the development and analysis of regularization methods, such as Tikhonov-type regularization methods or iterative regularization methods, which are a necessity in most of the inverse problems. In the last few decades, regularization methods motivating sparsity has been the focus of research, due to the high dimensionalty of the real-life data, and $\mathcal{L}^1$-regularization methods (such as LASSO or FISTA) has been in its center (due to their computational simplicity). In this paper we propose a new (semi-) iterative regularization method which is not only simpler than the mentioned algorithms but also yields better results, in terms of accuracy and sparsity of the recovered solution. Furthermore, we also present a very effective and practical stopping criterion to choose an appropriate regularization parameter (here, it's iteration index) so as to recover a regularized (sparse) solution. To illustrate the computational efficiency of this algorithm we apply it to numerically solve the image deblurring problem and compare our results with certain standard regularization methods, like total variation, FISTA, LSQR etc.