[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125247-en":3,"doc-seo-125247-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},125247,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Algebraic Frameworks and Computational Experiments for Enhanced Machine Learning - Thesis","Algebraic Frameworks and Computational Experiments for Enhanced Machine Learning presents methods to improve machine learning through structured, mathematical views of neural networks. The work develops algebraic representations for faster predictions in convolutional neural networks, investigates linear regions using tropical geometry, and supports these ideas with computational experiments. A dedicated chapter modifies U-Net for three-dimensional cancer segmentation, addressing practical challenges in training data, memory management, and class imbalance, and evaluating outcomes with clear future directions.","Algebraic Frameworks and Computational Experiments for Enhanced Machine Learning  \nby  \nJohnny Joyce  \nM.S. Artificial Intelligence, University of Southampton  \nB.S. Mathematics With a Year Abroad, Cardiff University  \nTHESIS  \nSubmitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the  \nUniversity of Illinois at Chicago, 2025  \nChicago, Illinois  \nDefense Committee: Jan Verschelde, Chair and Advisor Emily Dumas  \nGyörgy Turán Jie Yang  \nJonathan Gryak (City University of New York)  \nCopyright by  \nJohnny Joyce  \n2025  \nFor my incredible and inspiring wife, Aryanna Joyce.  \nI choose you every day and always.  \niii  \nACKNOWLEDGMENT  \nI thank my advisor, Professor Jan Verschelde, for his insights, guidance, and flexibility throughout my journey. From teaching me in his classes in the earlier years of my program to helping me navigate the world of research in the later years of my program, he has always given me great support.  \nI thank my each of my thesis committee members for guiding me in various ways: Professor György Turán for his thoughtful collaboration and discussions, and for guiding me throughout the making of Chapter 5; Professor Jonathan Gryak for having me to join him at his special session at the AMS JMM; Professor Emily Dumas for making my experience as a graduate teaching assistant exciting and rewarding through the multiple years we taught together; and Professor Jie Yang for teaching me in his classes on statistics.  \nI thank Li Lan of Clarix Imaging for producing dozens of high-quality training labels for Chapter 2, and I thank Doctor Xiao Han and Nikolaj Reiser of Clarix Imaging for their guidance and support throughout the medical imaging project in Chapter 2.  \nI thank Professor Luke Leisman and Professor Rick Laugesen of INMAS for providing thorough training and support that led to my internship with Clarix, making the work of Chapter 2 possible.  \nI thank my wife, Aryanna Joyce, for inspiring me with the kindness she brings to the world every day, and for always encouraging me to push through whatever difficulties arise.  \nACKNOWLEDGMENT (Continued)  \nI thank my mum, Lucy Joyce, for teaching me the values I hold, and for always helping me reach my full potential.  \nI thank my family across the UK, USA, Ireland, Philippines, Japan, and beyond, all of whom have made great impacts on my life even from afar.  \nI thank all my coworkers at iManage for supporting me and helping me grow professionally throughout the last two years.  \nJJ  \nCONTRIBUTIONS OF AUTHORS  \nChapter 3 is adapted from Joyce and Verschelde, 2024 [33], which was written by Johnny Joyce, and was edited, guided, and advised by Jan Verschelde.  \nChapter 5 is adapted from Chubarian et al., 2024 [13], which was co-authored by Karine Chubarian, Johnny Joyce, and György Turán. Johnny Joyce’s contributions included implementation, coding, experimentation, figure and table creation, and writeup of the experiments. The published version of this paper includes results of other authors, so the version in this thesis has been re-written and shortened to focus on contributions from Johnny Joyce.  \nTABLE OF CONTENTS  \nCHAPTER  PAGE  \n1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1  \n1.1 Problem Statement and Key Goals ................. 1  \n1.2 Description of Chapters and Contributions ............ 2  \n1.3 Scope .................................. 4  \n1.4 Overview of Relevant Background Literature ........... 5  \n1.5 Availability of Code .......................... 7  \n2 MODIFIED U-NET FOR THREE-DIMENSIONAL CANCER SEGMENTATION ............................... 8  \n2.1 Chapter Introduction and Goals .................. 8  \n2.2 3D Image Segmentation With Deep Learning .......... 11  \n2.2.1 Method Used: U-Net . . . . . . . . . . . . . . . . . . . . . . . . . 11  \n2.2.2 Challenges and Solutions ....................... 17  \n2.2.2.1 Obtaining Data for Trainin","cbCaiok4gCOhlu89","https://ap.wps.com/l/cbCaiok4gCOhlu89","pdf",10176327,1,169,"English","en",105,"# Introduction\n## Problem Statement and Key Goals\n## Description of Chapters and Contributions\n## Scope\n## Overview of Relevant Background Literature\n## Availability of Code\n# Modified U-Net for Three-Dimensional Cancer Segmentation\n## Chapter Introduction and Goals\n## 3D Image Segmentation With Deep Learning\n## Model Training and Evaluation\n## Outcomes and Future Directions\n# Algebraic Representations for Faster Predictions in Convolutional Neural Networks\n## Machine Learning and Algebraic Geometry\n## Convolutional Neural Network Setup\n## Loss Landscape Visualization\n## Computational Experiments\n## Future Directions\n# Exploring Linear Regions of Neural Networks with Tropical Geometry","[{\"question\":\"What main research theme ties the thesis together?\",\"answer\":\"The thesis connects machine learning performance improvements to algebraic structures, using mathematical representations and geometry to enable faster predictions and better understanding of neural network behavior.\"},{\"question\":\"How does the thesis address 3D cancer image segmentation?\",\"answer\":\"It presents a modified U-Net for three-dimensional cancer segmentation, focusing on training data preparation, memory management, and class imbalance, followed by model training and evaluation.\"},{\"question\":\"What role do computational experiments play?\",\"answer\":\"Computational experiments are used to validate ideas such as pre-computing convolutional neural networks and removing skip connections, with defined equipment, software, and data, plus practical experimentation and future directions.\"}]","Algebraic Frameworks and Computational Experiments for Enhanced Machine Learning - 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