基于分位数的迭代法求解受损线性方程组
Quantile-based Iterative Methods for Corrupted Systems of Linear Equations
摘要：
通常在从医学成像和传感器网络到纠错和数据科学（以及其他）的应用中，需要解决大规模线性系，其中一部分测量值已被破坏。我们考虑解决这种大型系统的线性方程$\mathbf{A}\mathbf{x}=\mathbf{b}$是不一致的，由于腐败的测量矢量$\mathbf{b}$。我们开发了几个变种的迭代法，收敛到未损坏的方程组系统的解决方案，即使在存在大的腐败。这些方法利用残差向量的绝对值的分位数来确定残差更新。我们提出的理论和实证结果表明，这些迭代方法的承诺。
英文摘要：
Often in applications ranging from medical imaging and sensor networks to error correction and data science (and beyond), one needs to solve large-scale linear systems in which a fraction of the measurements have been corrupted. We consider solving such large-scale systems of linear equations $\mathbf{A}\mathbf{x}=\mathbf{b}$ that are inconsistent due to corruptions in the measurement vector $\mathbf{b}$. We develop several variants of iterative methods that converge to the solution of the uncorrupted system of equations, even in the presence of large corruptions. These methods make use of a quantile of the absolute values of the residual vector in determining the iterate update. We present both theoretical and empirical results that demonstrate the promise of these iterative approaches.
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