Math 260L: Optimal transport
Professor: Katy Craig, katy•craig at ucsb • edu
Lecture: Monday and Wednesday, 10:30-11:45am, zoom (recordings will be posted below)
Office Hours: Monday and Wednesday, 11:45am-12:45pm, and by appointment, zoom
Optimal Transport Wiki: otwiki.xyz
Recommended References:
Exams: none.
Homework: Each student will write 2-3 articles for the Optimal Transport Wiki that we will create together. Potential topics are listed on the wiki. At least one article must be submitted by Friday, May 8th. Students will have the opportunity to revise submitted articles and will only be graded on the final version.
Here is a short video explaining how to create a new article on the wiki.
Grading Scheme: Homework: 10%, Surviving Covid-19: 90%
Prerequisites: Measure theory, functional analysis
Outline of Course:
| Part I: Optimal Transport | Part II: The Wasserstein Metric |
|---|---|
| Monge and Kantorovich Problems | approximation by convolution |
| convex analysis and duality in optimization | topology of Wasserstein metric |
| dual Kantorovich problem | Benamou-Brenier and dynamic characterization of Wasserstein metric |
| characterization of OT maps | Wasserstein geodesics and displacement interpolation |
A sample video on the topology of the Wasserstein metric and absolutely continuous curves (Lecture 16) is publicly available. The password to access to the remaining video is available by request.
Syllabus:
| topic | lecture notes | video | typed notes | ||
|---|---|---|---|---|---|
| 1 | Mar 30 (M) | general remarks | LEC1 | ||
| 2 | Apr 1 (W) | the Monge problem | LEC2 | VID2 | NOTE2 |
| 3 | Apr 6 (M) | from transport maps to transport plans | LEC3 | VID3 | NOTE3 |
| 4 | Apr 8 (W) | the Kantorovich problem | LEC4 | VID4 | NOTE4 |
| 5 | Apr 13 (M) | convergence of measures | LEC5 | VID5 | NOTE5 |
| 6 | Apr 15 (W) | convexity and the subdifferential | LEC6 | VID6 | NOTE6 |
| 7 | Apr 20 (M) | primal and dual optimization problems | LEC7 | VID7 | NOTE7 |
| 8 | Apr 22 (W) | the dual Kantorovich problem | LEC8 | VID8 | NOTE8 |
| 8 | Apr 27 (M) | equiv of primal and dual Kantorovich problems I | LEC9 | VID9 | NOTE9 |
| 10 | Apr 29 (W) | equiv of primal and dual Kantorovich problems II | LEC10 | VID10 | NOTE10 |
| 11 | May 1 (F) | optimal plans: the Knott-Smith criterion | LEC11 | VID11 | NOTE11 |
| 12 | May 6 (W) | optimal maps: Brenier's characterization | LEC12 | VID12 | NOTE12 |
| 13 | May 11 (M) | definition of Wasserstein metric | LEC13 | VID13 | NOTE13 |
| 14 | May 13 (W) | approximation of measures by convolution | LEC14 | VID14 | NOTE14 | 15 | May 18 (M) | Wasserstein metric: triangle inequality and topology | LEC15 | VID15 | NOTE15 |
| 16 | May 20 (W) | curves in the space of probability measures | LEC16 | VID16 | NOTE16 |
| 17 | May 27 (W) | duality again: Benamou Brenier | LEC17 | VID17 | NOTE17 |
| 18 | June 1 (M) | the continuity equation and Wasserstein geodesics | LEC18 | VID18 | NOTE18 |
| 20 | June 3 (M) | dynamic formulation of Wasserstein metric | LEC19 | VID19 | NOTE19 |