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.

中文翻译：

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线性方程组损坏的基于分位数的迭代方法

通常在从医学成像和传感器网络到纠错和数据科学（及其他）的应用中，人们需要解决其中一小部分测量已损坏的大规模线性系统。我们考虑求解此类大规模线性方程组 $\mathbf{A}\mathbf{x}=\mathbf{b}$ 由于测量向量 $\mathbf{b}$ 中的损坏而不一致。我们开发了几种迭代方法的变体，即使在存在大的损坏的情况下，它们也能收敛到未损坏的方程组的解。这些方法在确定迭代更新时利用残差向量的绝对值的分位数。我们提出了理论和实证结果，证明了这些迭代方法的前景。