On Subsampled Quantile Randomized Kaczmarz

When solving noisy linear systems Ax = b + c, the theoretical and empirical performance of stochastic iterative methods, such as the Randomized Kaczmarz algorithm, depends on the noise level. However, if there are a small number of highly corrupt measurements, one can instead use quantile-based methods to guarantee convergence to the solution x of the system, despite the presence of noise. Such methods require the computation of the entire residual vector, which may not be desirable or even feasible in some cases. In this work, we analyze the sub-sampled quantile Randomized Kaczmarz (sQRK) algorithm for solving large-scale linear systems which utilize a sub-sampled residual to approximate the quantile threshold. We prove that this method converges to the unique solution to the linear system and provide numerical experiments that support our theoretical findings. We additionally remark on the extremely small sample size case and demonstrate the importance of interplay between the choice of quantile and subset size

A Sampling Kaczmarz-Motzkin Algorithm for Linear Feasibility

We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. We obtain a family of algorithms that generalize and extend both projection-based techniques. We prove several convergence results, and our computational experiments show our algorithms often outperform the original methods

On Inferences from Completed Data

Matrix completion has become an extremely important technique as data scientists are routinely faced with large, incomplete datasets on which they wish to perform statistical inferences. We investigate how error introduced via matrix completion affects statistical inference. Furthermore, we prove recovery error bounds which depend upon the matrix recovery error for several common statistical inferences. We consider matrix recovery via nuclear norm minimization and a variant, $\ell_1$-regularized nuclear norm minimization for data with a structured sampling pattern. Finally, we run a series of numerical experiments on synthetic data and real patient surveys from MyLymeData, which illustrate the relationship between inference recovery error and matrix recovery error. These results indicate that exact matrix recovery is often not necessary to achieve small inference recovery error