# A Nonlinear Acceleration Method for Iterative Algorithms

## TL;DR

This paper introduces a nonlinear acceleration technique that enhances the convergence speed and stability of various iterative algorithms used in signal processing and sparse recovery tasks.

## Contribution

The paper proposes a novel nonlinear acceleration method that improves convergence and stability of multiple iterative algorithms, including IRLS, IMAT, SL0, ADMM, and Chebyshev Acceleration.

## Key findings

- Enhances convergence speed of iterative algorithms.
- Stabilizes and extends the stability range of existing methods.
- Applicable to a variety of algorithms in signal processing.

## Abstract

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.