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Structure, Symmetry, and Singularity in Learning Systems
Structure, Symmetry, and Singularity in Learning Systems
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20260202105644
- ISBN
- 9798270213121
- DDC
- 310
- 저자명
- Li, Jiayi.
- 서명/저자
- Structure, Symmetry, and Singularity in Learning Systems
- 발행사항
- [Sl] : University of California, Los Angeles, 2025
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2025
- 형태사항
- 224 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 87-06, Section: B.
- 주기사항
- Advisor: Montufar Cuartas, Guido F.
- 학위논문주기
- Thesis (Ph.D.)--University of California, Los Angeles, 2025.
- 초록/해제
- 요약We study structural, algebraic, and statistical aspects of learning systems, with an emphasis on understanding how structure, symmetry, and singularity influence their behavior. Part I examines models whose parameterizations or loss landscapes have polynomial or rational form. Using tools from algebraic geometry and numerical algebraic geometry, we investigate the geometry of their critical sets, the role of degeneracies and singularities, and the symmetries that arise from overparameterization or model design. This part also considers rational neural networks, for which we introduce algebraic regularization schemes aimed at improving trainability and analyze the resulting optimization landscapes through a combination of theoretical and numerical methods. Part II focuses on learning systems motivated by applications in statistics and engineering. Here we study procedures for estimating means of bounded random variables based on betting strategies, with an emphasis on statistical guarantees and practical performance. We also explore models of resilience and recovery in artificial and biological neural networks and investigate machine learning components used in digital twins for manufacturing systems, where structural assumptions play a central role in inference and control. The application of algebraic and numerical algebraic tools to machine learning theory is still at an early stage, and many questions remain open. A deeper understanding of how algebraic structure aligns with learning dynamics, how singularities arise in parameterized models, and how these features relate to implicit bias and other emergent phenomena may provide valuable insight. The mathematical ideas explored in this thesis reflect only a small part of what may be possible, but working with these tools has been a source of enjoyment due to the elegance they bring to complex systems. I hope that readers will find the methods and perspectives presented here accessible and motivating for future exploration.
- 일반주제명
- Statistics
- 일반주제명
- Industrial engineering
- 일반주제명
- Computer science
- 키워드
- Structure
- 키워드
- Symmetry
- 키워드
- Singularity
- 키워드
- Learning systems
- 키워드
- Machine learning
- 기타저자
- University of California, Los Angeles Statistics 0891
- 기본자료저록
- Dissertations Abstracts International. 87-06B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■020 ▼a9798270213121
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■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a310
■1001 ▼aLi, Jiayi.
■24510▼aStructure, Symmetry, and Singularity in Learning Systems
■260 ▼a[Sl]▼bUniversity of California, Los Angeles▼c2025
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2025
■300 ▼a224 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 87-06, Section: B.
■500 ▼aAdvisor: Montufar Cuartas, Guido F.
■5021 ▼aThesis (Ph.D.)--University of California, Los Angeles, 2025.
■520 ▼aWe study structural, algebraic, and statistical aspects of learning systems, with an emphasis on understanding how structure, symmetry, and singularity influence their behavior. Part I examines models whose parameterizations or loss landscapes have polynomial or rational form. Using tools from algebraic geometry and numerical algebraic geometry, we investigate the geometry of their critical sets, the role of degeneracies and singularities, and the symmetries that arise from overparameterization or model design. This part also considers rational neural networks, for which we introduce algebraic regularization schemes aimed at improving trainability and analyze the resulting optimization landscapes through a combination of theoretical and numerical methods. Part II focuses on learning systems motivated by applications in statistics and engineering. Here we study procedures for estimating means of bounded random variables based on betting strategies, with an emphasis on statistical guarantees and practical performance. We also explore models of resilience and recovery in artificial and biological neural networks and investigate machine learning components used in digital twins for manufacturing systems, where structural assumptions play a central role in inference and control. The application of algebraic and numerical algebraic tools to machine learning theory is still at an early stage, and many questions remain open. A deeper understanding of how algebraic structure aligns with learning dynamics, how singularities arise in parameterized models, and how these features relate to implicit bias and other emergent phenomena may provide valuable insight. The mathematical ideas explored in this thesis reflect only a small part of what may be possible, but working with these tools has been a source of enjoyment due to the elegance they bring to complex systems. I hope that readers will find the methods and perspectives presented here accessible and motivating for future exploration.
■590 ▼aSchool code: 0031.
■650 4▼aStatistics
■650 4▼aIndustrial engineering
■650 4▼aComputer science
■653 ▼aStructure
■653 ▼aSymmetry
■653 ▼aSingularity
■653 ▼aLearning systems
■653 ▼aMachine learning
■690 ▼a0463
■690 ▼a0984
■690 ▼a0800
■690 ▼a0546
■71020▼aUniversity of California, Los Angeles▼bStatistics 0891.
■7730 ▼tDissertations Abstracts International▼g87-06B.
■790 ▼a0031
■791 ▼aPh.D.
■792 ▼a2025
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17360956▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
