论文标题
用于稀疏正则化的梯度阈值算法
A Gradient-thresholding Algorithm for Sparse Regularization
论文作者
论文摘要
反问题在从工程到科学计算的领域中广泛应用中出现。与反对问题的兴趣兴趣有关的是正则化方法的开发和分析，例如Tikhonov型正则化方法或迭代正则化方法，这在大多数逆问题中都是必要的。在过去的几十年中，由于现实生活数据的高维度以及$ \ Mathcal {l}^1 $ regularization方法（例如Lasso或fista）（由于其计算简单性），激励稀疏性的正则化方法一直是研究的重点。在本文中，我们提出了一种新的（半）迭代正则化方法，该方法不仅比上述算法更简单，而且还可以在回收解决方案的准确性和稀疏性方面产生更好的结果。此外，我们还提出了一个非常有效且实用的停止标准，以选择适当的正则化参数（这里是迭代指数），以恢复正则化（稀疏）解决方案。为了说明该算法的计算效率，我们将其应用于数值解决图像脱张问题，并将结果与某些标准正则化方法进行比较，例如总变化，Fista，LSQR等。
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.
