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Type of Resource (primo): dissertations
Name: Personal
Name: Personal
Name: Personal
Name: Personal
Name: Corporate
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Note: thesis: Thesis (Ph. D.)--Brown University, 2022
Genre (aat): theses
Abstract: The growing computational demands of data science applications pose a significant challenge to machine learning and the applied sciences. These applications have relied mainly on increases in computing power to improve performance, but the computational power required to manage growing data sets and continue progress is soon expected to become economically and environmentally unsustainable. The design of efficient algorithms from problem domains (e.g., machine learning, imaging science, and optimal control) that take advantage of emerging hardware (e.g., field-programmable gate arrays architectures) has accordingly been identified as crucial to meet this challenge. Many traditional algorithms, however, were not developed to handle big data sets efficiently in this way. In this dissertation, I contribute innovative variational and Bayesian methods for large-scale machine learning and imaging science to try and meet this challenge. The focus is mathematical and supplemented with numerical examples. Chapter 2 of this dissertation introduces novel accelerated nonlinear primal-dual hybrid gradient methods tailored for efficiently solving a broad class of convex-concave saddle-point problems. I prove rigorous convergence results, including results for strongly convex or smooth problems posed on infinite-dimensional reflexive Banach spaces. Moreover, I establish novel connections between supervised learning tasks in machine learning and a broad class of first-order Hamilton--Jacobi partial differential equations with initial data. Chapter 3 of this dissertation applies the optimization methods developed in Chapter 2 to sparse logistic regression, regularized maximum entropy estimation, and entropy-regularized matrix games. I discuss each problem in detail, and I propose an explicit accelerated nonlinear primal-dual hybrid gradient method to solve each problem efficiently. I also present some numerical experiments to illustrate that my novel accelerated nonlinear primal-dual hybrid gradient methods are considerably faster than competing optimization methods. Finally, Chapter 4 of this dissertation presents new theoretical connections between a broad class of Bayesian posterior mean estimators for imaging science and viscous Hamilton--Jacobi partial differential equations with initial data. I use these connections to establish novel representation formulas and various properties of Bayesian posterior estimators.
Subject (fast) (authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01400410")
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