Venue: Chern Lecture Hall
Class Timings: Fridays from 9:00 AM - 11:00 AM (Hybrid Mode) and Mondays from 11:30 AM -1:00 PM (ICTS/TIFR-CAM/IISc students, will hold offline Q and A/discussion sessions )
First Meeting: 3 January 2025
Course Syllabus:
- Introduction to Monge-Kantorovich problem. Existence of solutions. Matchings and linear algebra
- Convex functions, Legendre transforms, convex conjugates
- Kantorovich duality
- Brenier’s Theorem, cyclical monotonicity
- 1-d transport, Knothe-Rosenblatt maps
- Stability Theorems
- Regularity of OT, entropy and relative entropy
- Entropic regularization of OT
- Peyré G, Cuturi M. Computational optimal transport: With applications to data science. Foundations and Trends® in Machine Learning. 2019 Feb 11;11(5-6):355-607.
- R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, 1997.
- F. Santambrogio, Optimal transport for applied mathematicians: Calculus of variations, pdes, and modeling, Progress in Nonlinear Differential Equations and Their Applications, Springer International Publishing, 2015.
- Cedric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, American Mathematical Society, 2003.
- T.M. Cover and J.A. Thomas, Elements of information theory, Wiley, 2006.
- Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, vol. 38, Springer, 1998.
- Giovanni Conforti and Luca Tamanini, A formula for the time derivative of the entropic cost and applications, J. Funct. Anal. 280 (2021), no. 1, 1–48.
- C. L ́eonard, A survey of the Schr ̈odinger problem and some of its connections with op- timal transport, Discrete Contin. Dyn. Syst. 34 (2014), no. 4, 1533–1574.
- T. Mikami, Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes, Probability Theory and Related Fields 129 (2004), no. 2, 245–260.
- S. Pal, On the difference between entropic cost and the optimal transport cost, Arxiv preprint, arxiv[math.PR] 1905.12206v2, 2019.
- L. Ru ̈schendorf and W. Thomsen, Note on the Schr ̈odinger equation and I-projections, Statistics and Probability Letters 17 (1993), 369–375.
Course Outcome: Student will be proficient in the fundamentals of optimal transport and will be able to explore research questions in the field.
- Teacher: Siva Athreya
Credit Score: 4