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Kaczmarz Methods and Structured Matr...

Yaniv, Yotam.

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Kaczmarz Methods and Structured Matrix Decompositions.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Kaczmarz Methods and Structured Matrix Decompositions./
Author:	Yaniv, Yotam.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2024,
Description:	133 p.
Notes:	Source: Dissertations Abstracts International, Volume: 85-11, Section: B.
Contained By:	Dissertations Abstracts International85-11B.
Subject:	Mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31297689
ISBN:	9798382599786

Kaczmarz Methods and Structured Matrix Decompositions.
Yaniv, Yotam.

Kaczmarz Methods and Structured Matrix Decompositions. - Ann Arbor : ProQuest Dissertations & Theses, 2024 - 133 p.

Source: Dissertations Abstracts International, Volume: 85-11, Section: B.

Thesis (Ph.D.)--University of California, Los Angeles, 2024.

In this dissertation, we discuss two distinct topics, both of which leverage randomized algorithms in numerical linear algebra. First we study three variants of the Kaczmarz method, a stochastic iterative method for solving linear systems. We propose a variant of the Kaczmarz method that uses additional memory to save on computation. We provide theoretical analysis and experimental results of the method, highlighting a gap in the literature. Additionally, we propose a variant of the Kaczmarz method in the data streaming setting that has an additional heavy ball momentum term. We prove a convergence bound for this method and analyze its merits experimentally given coherent data. Furthermore, we develop a variant of the Kaczmarz method for solving a latent class regression problem. Next we shift gears and discuss structured matrix factorizations. The first matrix factorization that we propose is a stratified non-negative matrix factorization. The aim of this method is to provide unsupervised dimensionality reduction on non-negative data that may be distributed across different locations. We prove a convergence bound for this method and analyze its performance on synthetic text, image and tabular data. Finally, we propose a hierarchically semi-separable matrix factorization method that uses random matrix sketching. 

ISBN: 9798382599786Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Linear algebra

Kaczmarz Methods and Structured Matrix Decompositions.
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100 1 $a Yaniv, Yotam. $3 3768169
245 1 0 $a Kaczmarz Methods and Structured Matrix Decompositions.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2024
300 $a 133 p.
500 $a Source: Dissertations Abstracts International, Volume: 85-11, Section: B.
500 $a Advisor: Bertozzi, Andrea;Hunter, Deanna M.
502 $a Thesis (Ph.D.)--University of California, Los Angeles, 2024.
520 $a In this dissertation, we discuss two distinct topics, both of which leverage randomized algorithms in numerical linear algebra. First we study three variants of the Kaczmarz method, a stochastic iterative method for solving linear systems. We propose a variant of the Kaczmarz method that uses additional memory to save on computation. We provide theoretical analysis and experimental results of the method, highlighting a gap in the literature. Additionally, we propose a variant of the Kaczmarz method in the data streaming setting that has an additional heavy ball momentum term. We prove a convergence bound for this method and analyze its merits experimentally given coherent data. Furthermore, we develop a variant of the Kaczmarz method for solving a latent class regression problem. Next we shift gears and discuss structured matrix factorizations. The first matrix factorization that we propose is a stratified non-negative matrix factorization. The aim of this method is to provide unsupervised dimensionality reduction on non-negative data that may be distributed across different locations. We prove a convergence bound for this method and analyze its performance on synthetic text, image and tabular data. Finally, we propose a hierarchically semi-separable matrix factorization method that uses random matrix sketching. 
590 $a School code: 0031.
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
653 $a Linear algebra
653 $a Matrix factorizations
653 $a Theoretical analysis
653 $a Kaczmarz method
653 $a Machine learning
690 $a 0405
690 $a 0642
690 $a 0800
690 $a 0364
710 2 $a University of California, Los Angeles. $b Mathematics 0540. $3 2096468
773 0 $t Dissertations Abstracts International $g 85-11B.
790 $a 0031
791 $a Ph.D.
792 $a 2024
793 $a English
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