Course Archives Theoretical Statistics and Mathematics Unit

Course: Special Topics - Analysis Instructor: Siva Athreya Room: G24 Level: Postgraduate Time: Currently offered

Syllabus Past Exams Syllabus: Pre-requisites:- Probablility Theory and Real analysis. 1. Introduction to Monge-Kantorovich problem. Existence of solutions. Matchings and linear algebra 2. Convex functions, Legendre transforms, convex conjugates 3. Kantorovich duality 4. Brenier’s Theorem, cyclical monotonicity 5. 1-d transport, Knothe-Rosenblatt maps 6. Stability Theorems 7. Regularity of OT, entropy and relative entropy 8. Entropic regularization of OT Suggested Texts : (a) 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. (b) R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, 1997. (c) 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. (d) C ́edric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, American Mathematical Society, 2003. (e) T.M. Cover and J.A. Thomas, Elements of information theory, Wiley, 2006. (f) Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, vol. 38, Springer, 1998. (g) 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. (h) 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. (i) 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. (j) S. Pal, On the difference between entropic cost and the optimal transport cost, Arxiv preprint, arxiv[math.PR] 1905.12206v2, 2019. (k) L. Ru ̈schendorf and W. Thomsen, Note on the Schr ̈odinger equation and I-projections, Statistics and Probability Letters 17 (1993), 369–375. Evaluation: Midterm Exam 25 marks Home Work / Assignment 25 marks Final Exam 50 marks Total 100 marks Top of the page Past Exams

Midterm 25.pdf Semestral Supplementary and Back Paper Top of the page

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