  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 QuantileRK's strength.
