Article revision ideas

Below, you can find a list of revision ideas for articles. If you choose to write about one of these ideas, remove it from the list below and email me.

Please do not revise your own article unless you check with me first.

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List of Articles

Monge Problem

• The definition of a transport map and the $t\#\mu =\nu$ notation should be highlighted, possibly in a separate section or definition environment.
• The equation expressing the Monge problem could be improved -- what is the infimum taken over? Why is the function F(T) defined within the infimum?
• There are errors in the formatting of the references.
• The "Issues with Formulation" section could be renamed "Challenges of Monge Problem". This section could be expended.
• The relation with the Kantorovich problem should be expanded. A citation to the wiki article no the Kantorovich article should be included. Explain precisely why the Kantorovich problem generalizes the Monge Problem.
• Add a section stating Brenier's theorem on solvability of the Monge problem and relation to Kantorovich's problem. Cite work by Gigli (see Santambrogio for the reference) on why $\mu$ absolutely continuous with respect to Lebesgue may be weakened to not giving mass on sets of d-1 dimensional Hausdorff measure.

Kantorovich Problem

• This page refers to the Monge problem too much. It should stand on its own. A separate section should be added that includes all the information about the relationship to the Monge problem.
• I don't think the dual problem should be mentioned in the introduction. This should be mentioned later. Perhaps the ``Introduction section should be removed. I don't think the Shipper's problem needs to be mentioned here.
• There are some grammatical errors that should be fixed.
• Intuition for the meaning of the transport plan should be given -- $\pi (A\times B)$ represents the amount of mass that is sent from A to B.
• The statement of the more problem could be improved. The formatting of this problem makes it a little hard to see that it is a minimization problem.
• The section on Kantorovich Duality should link to the wiki article on Kantorovich Duality.
• A section should be added explaining how the Kantorovich article can be solved by direct menthod of the Calculus of Variations.
• A section should be added stating the Knott-Smith optimality criterion for OT plans.

Optimal Transport in One Dimension

• Suggested changes TBD

Kantorovich Dual Problem (for general costs)

• This article should be merged with article on the Kantorovich Dual Problem for c= d^2 -- this should be a separate section in the article.
• Add a section on the dual problem for c=d and why this is a much simpler problem.
• The precise statement of the theorem should come first, and the heuristic interpretation in terms of the Shipper's problem should come second.
• There are some grammar problems that should be fixed.
• The section on ideas of the proof should refer to the wiki articles on Fenchel-Rockafellar Duality.

Discrete Optimal Transport

• This article should be rewritten in a way that is understandable to an undergraduate who has taken vector Calculus and linear algebra. It's fine to also mention more advanced topics, to connect it to other articles on the wiki, but the point of this article is that, in the discrete setting, it is just a vector calculus problem.
• Add a sentence to the beginning of the article about the intuitive idea of discrete measures, e.g. something like ``probability vectors, where the indices of the vector correspond to the amount of mass given to different locations in space
• The formatting for the statute of the problems could be improved with more displayed equations. The organization could be improved.
• Fewer parenthetical statements would be better.
• Better formatting could help make it more clear the notion of transport plan and map in this setting. It would be good to visually distinguish the Kantorovich and Monge Problem.
• What is the analogue of the dual Kantorovich problem in the discrete setting?
• When is the Monge problem solvable in the discrete setting? (We certainly won't have absolute continuity wrt Lebesgue, but it is still solvable in many cases...)
• The sentence in the section on Useful Combinatorial Structure could be a footnotes to a pervious section.
• The Algorithms section could be changed to just be the last sentence in the introduction.

Fenchel-Rockafellar and Linear Programming

• The Application to Linear Programs section should be expanded and better formatted. The main result here should be clearly stated (e.g. The dual of a linear program is XXX), and the rest of the argument should be formatted as the proof of this statement, via Fenchel-Rochafellar. The statement of the linear program should be made more conventional, see, e.g. [[1]]. The dual variable $\theta _{Y}$ should be introduced later.
• More should be added to the introduction on how the goal of this article is to show how the Fenchel-Rockafellar theorem can be used to characterize the dual of finite dimensional linear programs.

Optimal Transport and Ricci curvature

• Revision can begin on February 26
• Add a new section to the article discussing the relationship between Ricci curvature and convexity of the entropy. You should link to the existing wiki article on Geodesics and generalized geodesics.

Martingale optimal transport and mathematical finance

• Introduce some background material to explain the notation of what it means to be the expectation with respect to a filtration.
• Give intuition behind item (ii) in teh definition of an equivalent martingale measure
• Provide more intuition/background behind Problem 1 and 2. For example in Problem 2, what does it mean to be a martingale under Q? Why is that a natural requirement in this context?

Isoperimetric inequality and OMT

• Add sections describing recent work on quantitative isoperimetric inequalities, [2], [3]

Wasserstein barycenters and applications in image processing

• Revision can begin on February 26
• Suggested changes TBD

Wasserstein metric

• Revision can begin on February 26
• Suggested changes TBD