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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)
* Kantorovich Problem; Villani (1-3, 6-9), Santambrogio (xv-xvii,1-9)
* Kantorovich Dual Problem (for general costs); Villani (17-21), Santambrogio (9-16)
* 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)
* Optimal Transport and the Monge Ampère equation; Santambrogio (xvi, 54-57)
* Optimal Transport in One Dimension; Villani (73-78); Santambrogio (59-67)


==Numerical Methods for Optimal Transport==
==Variants of Optimal Transport Problems==  
* Discrete Optimal Transport; Villani (5), Santambrogio (235-237), Peyré Cuturi (7-12)
* 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]
* Auction Algorithm; Santambrogio (238-240); Peyré Cuturi (37-39, 52-56)
* Multi-marginal optimal transport and density functional theory
* 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==
==The 2-Wasserstein Metric==
* Machine Learning [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6024256/ Kolouri, et al, ''Optimal Mass Transport: Signal processing and machine-learning applications'']
* Benamou-Brenier dynamic characterization of W2; Santambrogio (187-198); Villani (238-249) (make sure to cite existing wiki article on [[Geodesics and generalized geodesics]])
* 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'']
* 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]


==Mathematical Foundations: Functional Analysis==
==Numerical Methods for Optimal Transport==
* The dual of <math> C_o(X) </math> vs. <math> C_b(X) </math>; Villani (39-43); Santambrogio (4); Rudin ''Real and Complex Analysis'' (127-132)
* Computing OT via Benamou-Brenier; Santambrogio (220-225); Peyré, Cuturi (102-108)
* [[Convergence of Measures and Metrizability]] (click on link for references and explanation)
* Wasserstein Barycenters; Santambrogio (215-218); Peyré, Cuturi (138-144)


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


==Fake Articles==
==Applications:==
* Cute puppies
* 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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