This paper studies Anderson acceleration (AA) for fixed-point methods ${x}^{(k+1)}=q({x}^{(k)})$. It provides the first proof that when the operator $q$ is linear and symmetric, AA improves the root-linear convergence factor over the fixed-point iterations. When $q$ is nonlinear, yet has a symmetric Jacobian at the solution, a slightly modified AA algorithm is proved to have an analogous root-linear convergence factor improvement over fixed-point iterations. Simulations verify our observations. Furthermore, experiments with different data models demonstrate AA is significantly superior to the standard fixed-point methods for Tyler's M-estimation.

翻译：本文研究固定点方法${x}^{(k+1)}=q({x}^{(k)})$的安德森加速（AA）。当算子$q$为线性且对称时，本文首次证明AA能改善固定点迭代的根线性收敛因子。当$q$为非线性但其解处的雅可比矩阵对称时，经轻微修改的AA算法被证明在固定点迭代基础上具有类似的根线性收敛因子改进效果。仿真结果验证了上述发现。此外，基于不同数据模型的实验表明，在Tyler's M估计中，AA显著优于标准固定点方法。