  Large-scale systems of linear equations arise in machine learning, medical
imaging, sensor networks, and in many areas of data science. When the scale of
the systems are extreme, it is common for a fraction of the data or
measurements to be corrupted. The Quantile Randomized Kaczmarz (QRK) method is
known to converge on large-scale systems of linear equations Ax=b that are
inconsistent due to static corruptions in the measurement vector b. We prove
that QRK converges even for systems corrupted by time-varying perturbations.
Additionally, we prove that QRK converges up to a convergence horizon on
systems affected by time-varying noise and demonstrate that the noise affects
only the convergence horizon of the method, and not the rate of convergence.
Finally, we utilize Markov's inequality to prove a lower bound on the
probability that the largest entries of the QRK residual reveal the
time-varying corruption in each iteration. We present numerical experiments
which illustrate our theoretical results.
