Timeline for The largest Wasserstein distance to uniform distribution among all probability distributions with uniform marginals
Current License: CC BY-SA 3.0
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| Mar 12, 2017 at 4:17 | comment | added | O. Richard | Thanks for your update. I just read the references and hope to get some idea. In fact, I think for $W_2$ we can explicitly compute the Wasserstein distance, and reduce the problem to a matrix problem. See my posted answer and let me know if it is correct. My main interest is using $\ell_1$ norm. Also note that the $p$ in $\ell_p$ is NOT the order of Wasserstein distance, but merely the metric of the space where the random variable takes value. I modified the question to make it clear. | |
| Mar 12, 2017 at 2:40 | history | edited | Henry.L | CC BY-SA 3.0 |
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| Mar 12, 2017 at 2:34 | comment | added | Henry.L | @O.Richard see my update in the last paragraph and the first paragraph(a serious typo there). | |
| Mar 12, 2017 at 2:34 | history | edited | Henry.L | CC BY-SA 3.0 |
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| Mar 12, 2017 at 2:29 | comment | added | Henry.L | (1)[Cuesta-Albertos et.al] discussed how to figuring out the shape of a copula using minimization of Wasserstein distance. (2)For general copula it is not solved, for $C_{unif}$ I do not know and want to know if you happen to figure out. (3) Circular nbd is a kind of nbd similar to $\epsilon$-neighborhood when the manifold has positive curvature, you can use $\epsilon$-net in some circumstances to replace it. See Example 3.4.2 in Differential Geometry of Curves and Surfaces: A Concise Guide by Victor Andreevich Toponogov, for example. | |
| Mar 12, 2017 at 2:14 | comment | added | O. Richard | Thanks for your answer. I have several questions: (1) Could you be more specific on which example in [Cuesta-Albertos et.al] for maximal Wasserstein distance? (2) For the "unsolved" general problem, do you mean it is still unsolved even if it is for uniform $C_{unif}$, or just for a general copula? (3) What is a circular neighborhood? | |
| Mar 11, 2017 at 23:45 | history | answered | Henry.L | CC BY-SA 3.0 |