Talk:Wasserstein metric

message: It seems that this concept was reinvented by computer scientists in the late 1980s, who have been calling it "Earth mover's distance" (EMD) ever since.  In 2001 it was pointed out that EMD is very similar, if not identical, to the Wasserstein metric.  Perhaps the two articles should be merged?  
: Cedric Villani (the fields medalist) in his 2009 seminal work on Optimal transport says that Wasserstein distance is still the dominant usage, but earth mover's distance is rapidly catching up. In the CS community EMD is clearly the dominant one. This merits two articles in Wikipedia, one with a Computer Science focus, one with a Mathematics focus. An average CS person will not want to read the Math page and possibly vice versa.  — Preceding unsigned comment added by 129.27.155.34 (talk) 11:59, 1 September 2011 (UTC)
replies: text: If that is the difference, then it is not clear from the EMD page. However continuous and discrete distributions are distinct enough to warrant separate articles.  Wqwt (talk) 23:18, 14 February 2022 (UTC)

text: I aggree with the reasoning of  129.27.155.34  and that it merits two separate articles. Additionally the article on the example is much longer than the one of the principal article subject, I think that's another reason for keeping both articles. Regards, Kmw2700 (talk) 12:28, 29 September 2015 (UTC)

text: I also agree with the reasoning of  129.27.155.34.  However with this reasoning the emphasis of *this* article must be changed to being more mature mathematically, reflecting the differing roles of the two articles.  A suggestion: The second paragraph of the introduction about the EMD should be removed, this is only the p=1 case and even then is misleading as the infimum might not be achieved.  It should be replaced by a brief explanation of the association with EMD. In general this article should be more about mathematical properties, while pointing to the EMD article for information concerning computation and applications. SymplectoJim (talk) 09:58, 17 March 2019 (UTC)
thread_title:  Earth mover's distance 

message: The Earth mover's distance article attributes the idea to Gaspard Monge in 1781.  Is the attribution legitimate? --Jorge Stolfi (talk) 19:17, 28 October 2008 (UTC)
thread_title:  Gaspard Monge 

message: The text is atrocious in that it is left as LaTeX.  It needs to be cleaned up and presented in a readable format. —The preceding unsigned comment was added by 71.132.145.120 (talk) 16:26, 19 March 2007 (UTC)
thread_title:  Conversion to mathematics format 

message: As far as I can see, the "corrections" made by Doetoe are wrong. I remove them. --Trigamma 08:09, 6 June 2007 (UTC)
thread_title:  Lip - Changes by [[User:Doetoe|Doetoe]] 

message: First, please understand that the talk pages are here for a reason.  Yes, Wikipedeans make mistakes sometimes, but the place to sort them out is the talk page, not the article text itself.  On a more mathematical note, the earth-mover's distance is indeed the L1 Wasserstein distance, also known as W1 and by many other names.  It is not the L2 Wasserstein distance (a.k.a. W2) because the cost of moving a bit of earth is proportional to the distance moved, not the square of the distance moved.  You are right that the distance in use on M might well be the Euclidean (\ell^2) distance (if M is a subset of Rn, but that doesn't make the earth-mover's distance on measures on M L2-ish. Sullivan.t.j (talk) 01:17, 14 November 2009 (UTC)
thread_title: Changes by 144.92.237.192

message: It looks like this test uses the Wasserstein metric with p=2, but neither article mentions the connection.  — Preceding unsigned comment added by 130.234.244.141 (talk) 09:10, 20 May 2017 (UTC)
thread_title:  Similar to Cramér–von Mises criterion 

message: In this equation, \gamma represents at the same time a function \gamma(x,y) \mapsto [0,\infty) (transport plan) and a measure on M \times M.
\iint c(x,y) \gamma(x,y) \, \mathrm{d} x \, \mathrm{d} y = \int c(x,y) \, \mathrm{d} \gamma(x,y)
I think the transport plan should be named something else (as it is in fact the Radon–Nikodym derivative of the measure \gamma) and the above equation more thoroughly explained.  — Preceding unsigned comment added by Johann Laconte (talk • contribs) 15:37, 28 January 2022 (UTC)
thread_title:  Notations in "Intuition and connection to optimal transport" section 

message: It would be interesting to know the joint probability distribution that minimises the objective function, but I guess it is the product of the two Gaussians? Or is there no analytical closed form? (not obvious to me from the cited paper) 178.197.207.95 (talk) 06:41, 26 April 2024 (UTC)
replies: text: It is given in equation (2) of the cited reference. (It is Gaussian, but not of product form. Essentially, one has to maximise the correlation between the two components.) Hairer (talk) 08:58, 28 April 2024 (UTC)
thread_title:  Normal distribution example