arXiv subjects

Jacob D. Moorman

Publications and source records attributed to Jacob D. Moorman.

3 recordsLinked to original sources

arXiv2022

Randomized Extended Kaczmarz is a Limit Point of Sketch-and-Project

The sketch-and-project (SAP) framework for solving systems of linear equations has unified the theory behind popular projective iterative methods such as randomized Kaczmarz, randomized coordinate descent, and variants thereof. The randomized extended Kaczmarz (REK) method is a popular extension of randomized Kaczmarz for solving inconsistent systems, which has not yet been shown to lie within the SAP framework. In this work we show that, in a certain sense, REK may be expressed as the limit point of a family of SAP methods, but we argue that it is unlikely that REK can be translated into a SAP method itself. We provide an extensive theoretical analysis of the family of methods comprising said limit, including convergence guarantees and further connections to REK. We follow this with an array of experiments demonstrating these methods and their connections in practice.

math.NA↗

arXiv2022

Selectable Set Randomized Kaczmarz

The Randomized Kaczmarz method (RK) is a stochastic iterative method for solving linear systems that has recently grown in popularity due to its speed and low memory requirement. Selectable Set Randomized Kaczmarz (SSRK) is an variant of RK that leverages existing information about the Kaczmarz iterate to identify an adaptive "selectable set" and thus yields an improved convergence guarantee. In this paper, we propose a general perspective for selectable set approaches and prove a convergence result for that framework. In addition, we define two specific selectable set sampling strategies that have competitive convergence guarantees to those of other variants of RK. One selectable set sampling strategy leverages information about the previous iterate, while the other leverages the orthogonality structure of the problem via the Gramian matrix. We complement our theoretical results with numerical experiments that compare our proposed rules with those existing in the literature.

math.NA↗

arXiv2020

Randomized Kaczmarz with Averaging

The randomized Kaczmarz (RK) method is an iterative method for approximating the least-squares solution of large linear systems of equations. The standard RK method uses sequential updates, making parallel computation difficult. Here, we study a parallel version of RK where a weighted average of independent updates is used. We analyze the convergence of RK with averaging and demonstrate its performance empirically. We show that as the number of threads increases, the rate of convergence improves and the convergence horizon for inconsistent systems decreases.

math.NA↗