一种用于迭代算法的非线性加速方法
A Nonlinear Acceleration Method for Iterative Algorithms
AI总结：
本文提出一种用于迭代算法的非线性加速方法，能提升IM、IRLS、IMAT、SL0和ADMM等稀疏恢复算法的性能，加速并稳定Chebyshev Acceleration方法，同时通过降低对自适应速率的敏感性扩展稳定性范围。
摘要：
迭代方法促进了对缺失采样、反卷积、逆系统、脉冲噪声和椒盐噪声去除等问题的更好理解和解决。然而，收敛速度和/或答案精度等挑战依然存在。为了改进现有的迭代算法，本文讨论了一种非线性方法。从不同方面对该方法进行了分析，包括其收敛性及其加速递归算法的能力。我们表明，该方法能够改进作为非均匀采样重建算法的 Iterative Method (IM)，以及一些迭代稀疏恢复算法，例如用于求解 LASSO 问题族（包括 Lasso 本身、Lasso-LSQR 和 group-Lasso）的 Iterative Reweighted Least Squares (IRLS)、Iterative Method with Adaptive Thresholding (IMAT)、Smoothed l0 (SL0) 和 Alternating Direction Method of Multipliers (ADMM)。它还能加速并稳定著名的 Chebyshev Acceleration (CA) 方法。此外，所提算法通过降低迭代算法对自适应速率变化的敏感性，扩展了其稳定性范围。
英文摘要：
Iterative methods have led to better understanding and solving problems such as missing sampling, deconvolution, inverse systems, impulsive and Salt and Pepper noise removal problems. However, the challenges such as the speed of convergence and or the accuracy of the answer still remain. In order to improve the existing iterative algorithms, a non-linear method is discussed in this paper. The mentioned method is analyzed from different aspects, including its convergence and its ability to accelerate recursive algorithms. We show that this method is capable of improving Iterative Method (IM) as a non-uniform sampling reconstruction algorithm and some iterative sparse recovery algorithms such as Iterative Reweighted Least Squares (IRLS), Iterative Method with Adaptive Thresholding (IMAT), Smoothed l0 (SL0) and Alternating Direction Method of Multipliers (ADMM) for solving LASSO problems family (including Lasso itself, Lasso-LSQR and group-Lasso). It is also capable of both accelerating and stabilizing the well-known Chebyshev Acceleration (CA) method. Furthermore, the proposed algorithm can extend the stability range by reducing the sensitivity of iterative algorithms to the changes of adaptation rate.
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