Sketch and Project: Randomized Iterative Methods for Linear Systems and Inverting Matrices
It provides a new theoretical framework for designing randomized algorithms in numerical linear algebra, potentially benefiting practitioners needing efficient solvers for large-scale problems.
This thesis develops a novel sketch-and-project framework for randomized iterative methods to solve linear systems and invert matrices, demonstrating that these methods are easier to analyze and often faster and more scalable than traditional approaches.
Probabilistic ideas and tools have recently begun to permeate into several fields where they had traditionally not played a major role, including fields such as numerical linear algebra and optimization. One of the key ways in which these ideas influence these fields is via the development and analysis of randomized algorithms for solving standard and new problems of these fields. Such methods are typically easier to analyze, and often lead to faster and/or more scalable and versatile methods in practice. This thesis explores the design and analysis of new randomized iterative methods for solving linear systems and inverting matrices. The methods are based on a novel sketch-and-project framework. By sketching we mean, to start with a difficult problem and then randomly generate a simple problem that contains all the solutions of the original problem. After sketching the problem, we calculate the next iterate by projecting our current iterate onto the solution space of the sketched problem.