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Metric and Tame Geometry in Optimization
Metric and Tame Geometry in Optimization
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20250211151314
- ISBN
- 9798382841472
- DDC
- 510
- 저자명
- Tian, Tonghua.
- 서명/저자
- Metric and Tame Geometry in Optimization
- 발행사항
- [Sl] : Cornell University, 2024
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2024
- 형태사항
- 199 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 85-12, Section: B.
- 주기사항
- Advisor: Lewis, Adrian.
- 학위논문주기
- Thesis (Ph.D.)--Cornell University, 2024.
- 초록/해제
- 요약Structure in optimization is traditionally studied through variational analysis. Structural results usually require convexity or other regularity conditions. However, contemporary optimization involves problems that lie beyond this traditional scope, especially in machine learning, both in Euclidean space and in non-Euclidean metric spaces. This thesis pursues theories for such problems that reveal underlying geometric structure and aid algorithm design and analysis.A central philosophy in the thesis is to highlight the slope as a tool for capturing first-order information of objectives. This engenders a subgradient-free development: subgradients are indirect in depicting structure and do not extend naturally to non-Euclidean settings. The development focuses on two ideas traditionally studied through subgradients.The first idea is identification: the phenomenon of a set being identified by certain convergent sequences in finite time. By re-interpreting identification via the slope, the thesis extends the idea from Euclidean space to general metric spaces, and establishes results on objective growth rates and necessary conditions for optimization on metric spaces. An extension of identification behavior from discrete-time sequences to continuous-time trajectories is also included.The second idea is the Kurdyka-Lojasiewicz (KL) inequality: a popular tool for convergence analysis in optimization. Defining the KL inequality via the slope reveals its connection with identification, and more importantly its impact on complexity analysis of first-order methods in metric spaces.The prevalence of identifiability and of objectives satisfying the KL inequality is established in the literature from semialgebraic (or, more generally, tame) geometry. Based on the same technique, the thesis presents two other results. The first characterizes the structure of conservative gradient fields, a notion of generalized gradients especially useful in analyzing deep learning methods. The second concerns the generic prevalence of partly smooth set-valued mappings, a type of structure known to be associated with identifiable sets. Furthermore, the thesis gives a short slope-based proof for the fundamental fact that semialgebraic functions possess the KL property.Finally, to unify and relax traditional structural assumptions such as convexity, smoothness, and their compositions, the thesis proposes a new type of implicit structure - smooth approximate convexity. The thesis shows the prevalence of such structure and demonstrates its applicability with a Riemannian optimization example.
- 일반주제명
- Theoretical mathematics
- 일반주제명
- Applied mathematics
- 키워드
- Euclidean space
- 키워드
- Tame geometry
- 키워드
- Metric spaces
- 기타저자
- Cornell University Operations Research and Information Engineering
- 기본자료저록
- Dissertations Abstracts International. 85-12B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■007cr#unu||||||||
■020 ▼a9798382841472
■035 ▼a(MiAaPQ)AAI31237872
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a510
■1001 ▼aTian, Tonghua.▼0(orcid)0009-0005-9708-1170
■24510▼aMetric and Tame Geometry in Optimization
■260 ▼a[Sl]▼bCornell University▼c2024
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2024
■300 ▼a199 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 85-12, Section: B.
■500 ▼aAdvisor: Lewis, Adrian.
■5021 ▼aThesis (Ph.D.)--Cornell University, 2024.
■520 ▼aStructure in optimization is traditionally studied through variational analysis. Structural results usually require convexity or other regularity conditions. However, contemporary optimization involves problems that lie beyond this traditional scope, especially in machine learning, both in Euclidean space and in non-Euclidean metric spaces. This thesis pursues theories for such problems that reveal underlying geometric structure and aid algorithm design and analysis.A central philosophy in the thesis is to highlight the slope as a tool for capturing first-order information of objectives. This engenders a subgradient-free development: subgradients are indirect in depicting structure and do not extend naturally to non-Euclidean settings. The development focuses on two ideas traditionally studied through subgradients.The first idea is identification: the phenomenon of a set being identified by certain convergent sequences in finite time. By re-interpreting identification via the slope, the thesis extends the idea from Euclidean space to general metric spaces, and establishes results on objective growth rates and necessary conditions for optimization on metric spaces. An extension of identification behavior from discrete-time sequences to continuous-time trajectories is also included.The second idea is the Kurdyka-Lojasiewicz (KL) inequality: a popular tool for convergence analysis in optimization. Defining the KL inequality via the slope reveals its connection with identification, and more importantly its impact on complexity analysis of first-order methods in metric spaces.The prevalence of identifiability and of objectives satisfying the KL inequality is established in the literature from semialgebraic (or, more generally, tame) geometry. Based on the same technique, the thesis presents two other results. The first characterizes the structure of conservative gradient fields, a notion of generalized gradients especially useful in analyzing deep learning methods. The second concerns the generic prevalence of partly smooth set-valued mappings, a type of structure known to be associated with identifiable sets. Furthermore, the thesis gives a short slope-based proof for the fundamental fact that semialgebraic functions possess the KL property.Finally, to unify and relax traditional structural assumptions such as convexity, smoothness, and their compositions, the thesis proposes a new type of implicit structure - smooth approximate convexity. The thesis shows the prevalence of such structure and demonstrates its applicability with a Riemannian optimization example.
■590 ▼aSchool code: 0058.
■650 4▼aTheoretical mathematics
■650 4▼aApplied mathematics
■653 ▼aEuclidean space
■653 ▼aTame geometry
■653 ▼aKurdyka-Lojasiewicz property
■653 ▼aRiemannian optimization
■653 ▼aMetric spaces
■690 ▼a0796
■690 ▼a0642
■690 ▼a0364
■71020▼aCornell University▼bOperations Research and Information Engineering.
■7730 ▼tDissertations Abstracts International▼g85-12B.
■790 ▼a0058
■791 ▼aPh.D.
■792 ▼a2024
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17161131▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
