Core Concepts
The Quantile Randomized Kaczmarz (QRK) method converges at least linearly in expectation up to a convergence horizon even when the linear system is perturbed by time-varying noise and corruption. The rate of convergence depends only on the corruption rate, while the convergence horizon depends on both the corruption rate and the time-varying noise.
Abstract
The paper considers solving highly overdetermined linear systems Ax = b that are perturbed by both time-varying noise n(k) and corruption c(k) in the measurement vector b(k) = b + n(k) + c(k). The authors analyze the convergence of the Quantile Randomized Kaczmarz (QRK) method in this setting.
Key highlights:
QRK is a variant of the Randomized Kaczmarz method that avoids updates corresponding to highly corrupted data by only updating if the sampled residual magnitude is less than a sufficient fraction of the residual magnitudes.
The authors prove that QRK converges at least linearly in expectation up to a convergence horizon, even in the presence of time-varying noise and corruption.
The rate of convergence depends only on parameters determined by the time-varying corruption, in particular the corruption rate.
The convergence horizon depends on both the corruption rate and the time-varying noise in the system.
The authors also provide a lower bound on the probability that the indices of the corrupted equations can be identified by examining the largest entries of the residual.
Numerical experiments are presented to illustrate the theoretical results.