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==The Optimal Transport Problem==
==The Optimal Transport Problem==
Unless otherwise specified, all topics are for general cost functions ''c(x,y)''.
 
* Monge Problem; Villani (3-4, 6-9), Santambrogio (xiv-xvii,1-9)
==Variants of Optimal Transport Problems==
* Kantorovich Problem; Villani (1-3, 6-9), Santambrogio (xv-xvii,1-9)
* Entropic optimal transport and the Schrödinger bridge problem [https://www.math.ucdavis.edu/~saito/data/acha.read.s19/leonard_survey-schroedinger-problem-optxport.pdf][https://www.math.columbia.edu/~mnutz/docs/EOT_lecture_notes.pdf]
* Kantorovich Dual Problem (for general costs); Villani (17-21), Santambrogio (9-16)
* Multi-marginal optimal transport and density functional theory
* Kantorovich Dual Problem (for <math> c(x,y) = d(x,y) </math> where <math> d </math> is a metric); Villani (34)
 
* Kantorovich Dual Problem (for <math> c(x,y) = d(x,y)^2 </math> where <math> d </math> is a metric); Santambrogio (16-18)
==The 2-Wasserstein Metric==
* Optimal Transport and the Monge Ampère equation; Santambrogio (xvi, 54-57)
* Benamou-Brenier dynamic characterization of W2; Santambrogio (187-198); Villani (238-249) (make sure to cite existing wiki article on [[Geodesics and generalized geodesics]])
* Displacement convexity; Santambrogio (249-251,271-276); Villani (150-154) (make sure to cite existing wiki article on [[Geodesics and generalized geodesics]])
* 2-Wasserstein metric and Ricci curvature; Figalli-Glaudo (112-114); [https://cedricvillani.org/sites/dev/files/old_images/2012/08/P12.CIME_.pdf]


==Numerical Methods for Optimal Transport==
==Numerical Methods for Optimal Transport==
* Discrete Optimal Transport; Villani (5), Santambrogio (235-237), Peyré Cuturi (7-12)
* Computing OT via Benamou-Brenier; Santambrogio (220-225); Peyré, Cuturi (102-108)
* Auction Algorithm; Santambrogio (238-240); Peyré Cuturi (37-39, 52-56)
* Wasserstein Barycenters; Santambrogio (215-218); Peyré, Cuturi (138-144)
* Entropic Regularization; Santambrogio (240-241); Peyré Cuturi (57-62)
* Sinkhorn's Algorithm; Peyré Cuturi (62-73)
* Semidiscrete Optimal Transport (for <math> c(x,y) = |x-y|^2 </math>); Santambrogio (242-248); Peyré Cuturi (85-89)
 
==Applications of Optimal Transport==
* Machine Learning [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6024256/ Kolouri, et al, ''Optimal Mass Transport: Signal processing and machine-learning applications'']
* Economic Matching Problems; Santambrogio (44-48) [https://academic-oup-com.proxy.library.ucsb.edu:9443/ectj/article/20/2/C1/5051096 Galichon, ''A survey of some recent applications of optimal transport methods to econometrics'']
* Collider Events
* Optimal Control
* ...?


==Mathematical Foundations==
==Mathematical Foundations:==
* [Convergence of Measures and Metrizability] (click on link for references and explanation) Ambrosio, Gigli, Savaré (107-108), Brezis (72-76)
* Gradient flows in metric spaces; Figalli-Glaudo (107-108); Santambrogio, 'OT for Applied Mathematicians' (285-290); Santambrogio, 'Euclidean, Metric, and Wasserstein GFs' (90-107; don't need to cover all topics, just what interests you)


I'm hoping graduate students in other areas of science may write an article about how they are using optimal transport :). If you would like me to suggest references, let me know!
==Applications:==
* Wasserstein Generative Adversarial Networks [https://leon.bottou.org/publications/pdf/icml-2017.pdf], [https://nemo.kiwi/studies/M1/ML/report.pdf]
* Optimal transport methods in economics; see introduction of book by Galichon (I have a copy you can borrow) and [https://arxiv.org/abs/2107.04700]
* Quantization and Lloyd's algorithm [https://hal.archives-ouvertes.fr/hal-03256039/document], [https://link.springer.com/content/pdf/10.1007/978-3-030-01947-1_7.pdf], [https://link.springer.com/content/pdf/10.1007/978-3-319-99689-9_6.pdf]

Latest revision as of 07:46, 7 February 2022

Below, you can find a list of new article ideas and suggested references. (Feel free to incorporate additional references! Please list all references you use at the bottom of your article.) If you choose to write about one of these ideas, remove it from the list below and create a new link on the main page.

Want to write about something that's not listed here? Email me!

The Optimal Transport Problem

Variants of Optimal Transport Problems

  • Entropic optimal transport and the Schrödinger bridge problem [1][2]
  • Multi-marginal optimal transport and density functional theory

The 2-Wasserstein Metric

  • Benamou-Brenier dynamic characterization of W2; Santambrogio (187-198); Villani (238-249) (make sure to cite existing wiki article on Geodesics and generalized geodesics)
  • Displacement convexity; Santambrogio (249-251,271-276); Villani (150-154) (make sure to cite existing wiki article on Geodesics and generalized geodesics)
  • 2-Wasserstein metric and Ricci curvature; Figalli-Glaudo (112-114); [3]

Numerical Methods for Optimal Transport

  • Computing OT via Benamou-Brenier; Santambrogio (220-225); Peyré, Cuturi (102-108)
  • Wasserstein Barycenters; Santambrogio (215-218); Peyré, Cuturi (138-144)

Mathematical Foundations:

  • Gradient flows in metric spaces; Figalli-Glaudo (107-108); Santambrogio, 'OT for Applied Mathematicians' (285-290); Santambrogio, 'Euclidean, Metric, and Wasserstein GFs' (90-107; don't need to cover all topics, just what interests you)

Applications:

  • Wasserstein Generative Adversarial Networks [4], [5]
  • Optimal transport methods in economics; see introduction of book by Galichon (I have a copy you can borrow) and [6]
  • Quantization and Lloyd's algorithm [7], [8], [9]
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