Concetti Chiave
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.
Sintesi
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.