UCLA

In this thesis, we study variants of Kaczmarz's iterative method for solving systems of linear equations. We introduce and analyze a variant that incorporates heavy-ball momentum to accelerate convergence when the data is highly coherent. We provide theoretical and empirical results for both fixed systems and streamed data. Furthermore, we introduce variants of the method tailored to a specific sparse corruption model. These variants leverage information from the residual to avoid corrupted rows and estimate the solution's direction. We offer theoretical guarantees and empirical evidence showcasing the convergence of our methods to the uncorrupted system's solution. Additionally, we establish an equivalence between block variants of Kaczmarz's method and gossip protocols, which are widely used in distributed computing. We develop new convergence theory on both sides of this equivalence and extend our results to account for noisy models of network communication.