Quantile-Based Random Kaczmarz for corrupted linear systems of equations

Stefan Steinerberger

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arXiv · 2107.05554

Quantile-Based Random Kaczmarz for corrupted linear systems of equations

Stefan Steinerberger

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Abstract

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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Stefan Steinerberger. 2021-07-12. Quantile-Based Random Kaczmarz for corrupted linear systems of equations. https://arxiv.org/abs/2107.05554

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