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Measurement data in linear systems arising from real-world applications often suffers from both large, sparse corruptions, and widespread small-scale noise. This can render many popular solvers ineffective, as the least squares solution is far from the desired solution, and the underlying consistent system becomes harder to identify and solve. QuantileRK is a member of the Kaczmarz family of iterative projective methods that has been shown to converge exponentially for systems with arbitrarily large sparse corruptions. In this paper, we extend the analysis to the case where there are not only corruptions present, but also noise that may affect every data point, and prove that QuantileRK converges with the same rate up to an error threshold. We give both theoretical and experimental results demonstrating QuantileRKs strength.
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face difficult
In this paper, combining count sketch and maximal weighted residual Kaczmarz method, we propose a fast randomized algorithm for large overdetermined linear systems. Convergence analysis of the new algorithm is provided. Numerical experiments show tha
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 $beta m$ entries of $b$ have been corrupted (possibly by arbitrarily large numbers). Haddock, Needell, Rebrova an
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 s
Based on the geometric {it Triangle Algorithm} for testing membership of a point in a convex set, we present a novel iterative algorithm for testing the solvability of a real linear system $Ax=b$, where $A$ is an $m times n$ matrix of arbitrary rank.