Quantile-based random sparse Kaczmarz for corrupted and noisy linear systems

The randomized Kaczmarz method, along with its recently developed variants, has become a popular tool for dealing with large-scale linear systems. However, these methods usually fail to converge when the linear systems are affected by heavy corruption, which is common in many practical applications. In this study, we develop a new variant of the randomized sparse Kaczmarz method with linear convergence guarantees, by making use of the quantile technique to detect corruptions. Moreover, we incorporate the averaged block technique into the proposed method to achieve parallel computation and acceleration. Finally, the proposed algorithms are illustrated to be very efficient through extensive numerical experiments.