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Statistical and Algorithmic Perspectives on Estimating Optimal Transport Maps
Statistical and Algorithmic Perspectives on Estimating Optimal Transport Maps
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20260202104704
- ISBN
- 9798293886982
- DDC
- 519
- 서명/저자
- Statistical and Algorithmic Perspectives on Estimating Optimal Transport Maps
- 발행사항
- [Sl] : New York University, 2025
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2025
- 형태사항
- 318 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 87-03, Section: B.
- 주기사항
- Advisor: Niles-Weed, Jonthan.
- 학위논문주기
- Thesis (Ph.D.)--New York University, 2025.
- 초록/해제
- 요약Optimal transport maps, or Brenier maps, have become widely adopted in data-driven domains as they provide a canonical transformation between independent datasets. While many existing methods aim for optimal statistical performance, they often fall short in practical regimes of interest, such as when the data is high-dimensional or when the sample size is large. In this thesis, we analyze principled algorithms for estimating optimal transport maps in precisely these regimes. Our rst contribution is the introduction of the entropic Brenier map, an estimator of the Brenier map based on entropic optimal transport, which harnesses the computational e ciency of Sinkhorn's matrix scaling algorithm (Sinkhorn, 1967). We prove the rst nitesample guarantees for estimating optimal transport maps using this estimator, demonstrate that it is minimax optimal in the semi-discrete setting, and make further connections to the statistical estimation of Schrodinger bridge between two distributions. Next, we further derive new theoretical properties of the entropic Brenier map, such as bounds on the Lipschitz constant of the map as well as its stability with respect to the target measures; these results also yield new insights for the unregularized optimal transport map. For our nal contribution, we propose a new optimization framework for functionals dened over a suitable family of optimal transport maps. As an application, we develop the rst gradient-based algorithm for mean-eld variational inference that comes with end-to-end convergence guarantees.
- 일반주제명
- Applied mathematics
- 일반주제명
- Statistics
- 기타저자
- New York University Center for Data Science
- 기본자료저록
- Dissertations Abstracts International. 87-03B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■006m o d
■007cr#unu||||||||
■020 ▼a9798293886982
■035 ▼a(MiAaPQ)AAI32116765
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a519
■1001 ▼aPooladian, Aram-Alexandre.
■24510▼aStatistical and Algorithmic Perspectives on Estimating Optimal Transport Maps
■260 ▼a[Sl]▼bNew York University▼c2025
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2025
■300 ▼a318 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 87-03, Section: B.
■500 ▼aAdvisor: Niles-Weed, Jonthan.
■5021 ▼aThesis (Ph.D.)--New York University, 2025.
■520 ▼aOptimal transport maps, or Brenier maps, have become widely adopted in data-driven domains as they provide a canonical transformation between independent datasets. While many existing methods aim for optimal statistical performance, they often fall short in practical regimes of interest, such as when the data is high-dimensional or when the sample size is large. In this thesis, we analyze principled algorithms for estimating optimal transport maps in precisely these regimes. Our rst contribution is the introduction of the entropic Brenier map, an estimator of the Brenier map based on entropic optimal transport, which harnesses the computational e ciency of Sinkhorn's matrix scaling algorithm (Sinkhorn, 1967). We prove the rst nitesample guarantees for estimating optimal transport maps using this estimator, demonstrate that it is minimax optimal in the semi-discrete setting, and make further connections to the statistical estimation of Schrodinger bridge between two distributions. Next, we further derive new theoretical properties of the entropic Brenier map, such as bounds on the Lipschitz constant of the map as well as its stability with respect to the target measures; these results also yield new insights for the unregularized optimal transport map. For our nal contribution, we propose a new optimization framework for functionals dened over a suitable family of optimal transport maps. As an application, we develop the rst gradient-based algorithm for mean-eld variational inference that comes with end-to-end convergence guarantees.
■590 ▼aSchool code: 0146.
■650 4▼aApplied mathematics
■650 4▼aStatistics
■653 ▼aEntropic regularization
■653 ▼aNon-parametric statistics
■653 ▼aOptimal transport
■653 ▼aVariational inference
■690 ▼a0364
■690 ▼a0463
■71020▼aNew York University▼bCenter for Data Science.
■7730 ▼tDissertations Abstracts International▼g87-03B.
■790 ▼a0146
■791 ▼aPh.D.
■792 ▼a2025
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17358451▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
