  The randomzied 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 corruptions, which are common in many practical applications. In this
study, we develop a new variant of the randomzied sparse Kaczmarz method with
linear convergence guarantees, by making use of a quantile technique to detect
corruptions. Moreover, we incorporate 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.
