The Kaczmarz Algorithm and doubly noisy linear systems

Large-scale linear systems, Ax=b, frequently arise in data science and scientific computing at massive scales, thus demanding effective iterative methods to solve them. Often, these systems are noisy due to operational errors or faulty data-collection processes. In the past decade, the randomized Kaczmarz algorithm (RK) was studied extensively as an efficient iterative solver for such systems. However, the convergence study of RK in the noisy regime is limited and only considers measurement noise in the right-hand side vector, b. Practically, that is not always the case, as the coefficient matrix A can also be noisy. But is noise inherently bad? In this talk, we motivate and discuss doubly noise linear systems and the performance of the Kaczmarz-type algorithms applied to such systems, highlighting that noise can potentially be exploited. The presented work is a joint work with El Houcine Bergou, Soumia Boucherouite, Aritra Dutta, and Xin Li.

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