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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)''.


* Kantorovich Dual Problem (for general costs); Villani (17-21), Santambrogio (9-16)
==Variants of Optimal Transport Problems==
* Kantorovich Dual Problem (for <math> c(x,y) = d(x,y) </math> where <math> d </math> is a metric); Villani (34)
* 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 <math> c(x,y) = d(x,y)^2 </math> where <math> d </math> is a metric); Santambrogio (16-18)
* Multi-marginal optimal transport and density functional theory
* Optimal Transport in One Dimension; Villani (73-78); Santambrogio (59-67)
 
==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); [https://cedricvillani.org/sites/dev/files/old_images/2012/08/P12.CIME_.pdf]


==Numerical Methods for Optimal Transport==
==Numerical Methods for Optimal Transport==
* Entropic Regularization; Santambrogio (240-241); Peyré Cuturi (57-62)
* Computing OT via Benamou-Brenier; Santambrogio (220-225); Peyré, Cuturi (102-108)
* Sinkhorn's Algorithm; Peyré Cuturi (62-73)
* Wasserstein Barycenters; Santambrogio (215-218); Peyré, Cuturi (138-144)
* 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'']


==Mathematical Foundations: Functional Analysis==
==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)


==Mathematical Foundations: Optimization==
==Applications:==
* Fenchel-Rockafellar and Linear Programming; Brezis (15-17); Rockafellar, ''Variational Analysis'' (505-507)
* 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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