损坏线性系的基于分位数的随机Kaczmarz算法
Quantile-Based Random Kaczmarz for corrupted linear systems of equations
摘要：
我们考虑线性系$Ax = b$，其中$A \in \mathbb{R}^{m \times n}$由规范化的行，$\|a_i\|_{\ell^2} = 1$，其中高达$βm$条目的$b$已损坏（可能由任意大的数字）。Haddock、Needell、Rebrova和Swartworth提出了一种基于分位数的随机Kaczmarz方法，并证明了对于某些随机矩阵$A$，它以很高的似然收敛到真实解。我们证明了一个确定性的版本，任何矩阵$A$，$β_A$的数量，使有收敛的所有扰动与$β< β_A$。假设随机矩阵启发式，这证明了具有高达$\sim 0.5\%$损坏（可能会改进的数字）的高高斯矩阵的收敛性。
英文摘要：
We consider linear systems $Ax = b$ where $A \in \mathbb{R}^{m \times n}$ consists of normalized rows, $\|a_i\|_{\ell^2} = 1$, and where up to $βm$ entries of $b$ have been corrupted (possibly by arbitrarily large numbers). Haddock, Needell, Rebrova and Swartworth propose a quantile-based Random Kaczmarz method and show that for certain random matrices $A$ it converges with high likelihood to the true solution. We prove a deterministic version by constructing, for any matrix $A$, a number $β_A$ such that there is convergence for all perturbations with $β< β_A$. Assuming a random matrix heuristic, this proves convergence for tall Gaussian matrices with up to $\sim 0.5\%$ corruption (a number that can likely be improved).
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