Optimal Transport I
This course is meant to be a gentle introduction to optimal transport and will be the first part in a two-part course. In this course, we will discuss in detail the discrete Monge and Kantorovich problems and address various key points such as existence and uniqueness of solutions of such problems. We will also have a brief introduction to convex analysis and convex optimization in order to better understand the material. Then, we will delve into computational methods for discrete optimal transport such as the simplex method and the Sinkhorn algorithm arising from the entropically regularized optimal transport problem. Lastly, if time permits, we will briefly discuss explicit formulas for some optimal transport problems on the real line and finish with an introduction to the semi-discrete optimal transport problem.
日期
2025年09月16日 至 12月11日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周二,周四 | 15:20 - 16:55 | - | - | - |
修课要求
Real analysis, linear algebra, basic measure theory, some coding experience
课程大纲
Week 1: Introduction to discrete Monge problem and introduction to discrete Kantorovich problem, first observations regarding existence issues for Monge problem uniqueness issues for Kantorovich problem
Week 2: Metric properties of (discrete) Wasserstein distance, convex analysis introduction
Week 3: Convex analysis introduction
Week 4: Convex optimization problems, introduction to linear programming
Week 5: Geometry of optimal transport for squared distance cost function, characterizing the minimizers of the Monge and Kantorovich problems
Week 6: Geometry of optimal transport for squared distance cost function, characterizing the minimizers of the Monge and Kantorovich problems
Week 7: Dual formulation of convex optimization problems and optimal transport problem, KKT conditions
Week 8: Discussion about other cost functions, introduction to numerical methods for optimal transport
Week 9: Simplex method for solving linear programming problems
Week 10: Entropic regularization of optimal transport, Sinkhorn algorithm
Week 11: Optimal transport on the real line
Week 12: Semi-discrete optimal transport
Week 2: Metric properties of (discrete) Wasserstein distance, convex analysis introduction
Week 3: Convex analysis introduction
Week 4: Convex optimization problems, introduction to linear programming
Week 5: Geometry of optimal transport for squared distance cost function, characterizing the minimizers of the Monge and Kantorovich problems
Week 6: Geometry of optimal transport for squared distance cost function, characterizing the minimizers of the Monge and Kantorovich problems
Week 7: Dual formulation of convex optimization problems and optimal transport problem, KKT conditions
Week 8: Discussion about other cost functions, introduction to numerical methods for optimal transport
Week 9: Simplex method for solving linear programming problems
Week 10: Entropic regularization of optimal transport, Sinkhorn algorithm
Week 11: Optimal transport on the real line
Week 12: Semi-discrete optimal transport
参考资料
1. Optimal Mass Transport on Euclidean Spaces by Maggi
2. Convex Analysis by Rockafellar
3. Convex Optimization by Boyd and Vanderberghe
4. Numerical Optimization by Nocedal and Wright
5. Computational Optimal Transport by Peyré and Cuturi
2. Convex Analysis by Rockafellar
3. Convex Optimization by Boyd and Vanderberghe
4. Numerical Optimization by Nocedal and Wright
5. Computational Optimal Transport by Peyré and Cuturi
听众
Advanced Undergraduate
, Graduate
, 博士后
, Researcher
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笔记公开
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语言
英文