This monograph aims to offer a concise introduction to optimal transport, quickly transitioning to its applications in statistics and machine learning. It is primarily tailored for students and researchers in these fields, yet it remains accessible to a broader audience of applied mathematicians and computer scientists. Each chapter is complemented with exercises for the reader to test their understanding. As such, this monograph is suitable for a graduate course on the topic of statistical optimal transport.

Gathers scattered material on optimal transport central to applications in statistics and machine learning Provides a unique overview of the current state of statistical optimal transport Includes in-depth mathematical treatment of novel questions arising in statistical optimal transport

Autorentext

Sinho Chewi is an Assistant Professor of Statistics and Data Science at Yale University. He obtained his PhD in Mathematics and Statistics from the Massachusetts Institute of Technology in 2023, under the supervision of Philippe Rigollet. He works broadly on the mathematics of machine learning and statistics, with a focus on applications of optimal transport to computational problems arising in those fields. He is currently writing a book on log-concave sampling.

Jonathan Niles-Weed is an Associate Professor of Mathematics and Data Science at New York University. He studies mathematical statistics, the mathematics of data science, and applications of optimal transport in statistics, probability, and machine learning. He holds a PhD from the Massachusetts Institute of Technology and is the recipient of a Sloan Fellowship in Mathematics, an NSF CAREER award, the 2023 Tweedie New Researcher Award from the Institute for Mathematical Statistics, and the 2024 Early Career Prize from the SIAM Activity Group on Data Science.

Philippe Rigollet is the Cecil and Ida Green Distinguished Professor of Mathematics at MIT, where he serves as Chair of the Applied Mathematics Committee. He works at the intersection of statistics, machine learning, and optimization, focusing primarily on the design and analysis of efficient statistical methods. His current research is on statistical optimal transport and the mathematical theory behind transformers. His research has been recognized by the CAREER award from the National Science Foundation and a Best Paper Award at the Conference on Learning Theory in 2013 for his pioneering work on statistical-to-computational tradeoffs. He is an elected fellow of the Institute of Mathematical Statistics and gave a Medallion lecture at the Joint Statistical Meetings in 2021.

Inhalt

1. Optimal Transport.- 2. Estimation of Wasserstein distances.- 3. Estimation of transport maps.- 4. Entropic optimal transport.- 5. Wasserstein gradient flows: theory.-6. Wasserstein gradient flows: applications.- 7. Metric geometry of the Wasserstein space.- 8. Wasserstein barycenters.