A group of projection based approaches for solving large-scale linear systems is known for its speed and simplicity. For example, Kaczmarz algorithm iteratively projects the previous approximation x_k onto the solution spaces of the next equation in the system. An elegant proof of the exponential convergence of this method, using correct randomization of the process, was given in 2009 by Strohmer and Vershynin, and succeeded by many extensions and generalizations. I will discuss our newly developed variants of these methods that successfully avoid large and potentially adversarial corruptions in the linear system. I specifically focus on the random matrix and high-dimensional probability results that play a crucial role in proving convergence of such methods. Based on the joint work with Jamie Haddock, Deanna Needell, and Will Swartworth.
