Timeline for The largest Wasserstein distance to uniform distribution among all probability distributions with uniform marginals
Current License: CC BY-SA 3.0
31 events
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| Feb 27, 2020 at 10:51 | comment | added | Gilles Mordant | Would you mind pointing where the proof for the $W_2$ distance is proposed ? I read the entire thread and could not locate it. | |
| Mar 13, 2017 at 3:13 | vote | accept | O. Richard | ||
| Mar 13, 2017 at 0:37 | history | edited | O. Richard | CC BY-SA 3.0 |
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| Mar 12, 2017 at 22:12 | answer | added | Steve | timeline score: 2 | |
| Mar 12, 2017 at 18:52 | history | edited | O. Richard | CC BY-SA 3.0 |
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| Mar 12, 2017 at 4:20 | history | edited | O. Richard | CC BY-SA 3.0 |
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| S Mar 12, 2017 at 0:12 | history | suggested | Henry.L |
it is not about "matrices"
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| Mar 11, 2017 at 23:52 | review | Suggested edits | |||
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| Mar 11, 2017 at 23:45 | answer | added | Henry.L | timeline score: 2 | |
| Mar 11, 2017 at 20:31 | history | edited | O. Richard | CC BY-SA 3.0 |
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| Mar 11, 2017 at 14:42 | history | edited | O. Richard | CC BY-SA 3.0 |
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| Mar 11, 2017 at 4:57 | history | edited | O. Richard | CC BY-SA 3.0 |
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| Mar 10, 2017 at 20:14 | history | edited | O. Richard | CC BY-SA 3.0 |
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| Mar 10, 2017 at 20:13 | comment | added | O. Richard | @TomSolberg Mainly of theoretical interest, I don't know a particular application yet... But I think this problem may occur in other fields without in the language of Wasserstein distance. | |
| Mar 10, 2017 at 18:28 | comment | added | Tom Solberg | Just out of curiosity, what motivated you to study this problem? Is there a practical application? | |
| Mar 10, 2017 at 17:18 | comment | added | O. Richard | @BjørnKjos-Hanssen Yeah, this intuition makes a lot of sense. However, I cannot build a RIGOROUS connection between the goal of the problem and the goal of find a distribution with smallest neighborhood... | |
| Mar 10, 2017 at 16:09 | comment | added | Bjørn Kjos-Hanssen | The graph of $j=i+N/2$ has a larger "neighborhood". I wonder if $i=j$ and $i+j=N+1$, being straight lines, have the smallest neighborhood by some "isoperimetric" theorem? | |
| Mar 10, 2017 at 15:11 | history | edited | O. Richard | CC BY-SA 3.0 |
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| Mar 10, 2017 at 15:05 | comment | added | O. Richard | @BenoîtKloeckner The dyadic squares may be a good starting point for proof though. | |
| Mar 10, 2017 at 14:58 | comment | added | O. Richard | @BenoîtKloeckner I just check $N=4$ by solving the linear programming that defines Wasserstein distance. The distance between comonotonic distribution and $\nu$ is 0.25, whereas the distance between uniform distribution on $j=i+N/2 \mod N$ and $\nu$ is 0.2. | |
| Mar 10, 2017 at 13:21 | comment | added | Benoît Kloeckner | Sorry, I misread your question. I would still check whether $\mu$ uniform on $(j=i+N/2 \mod N)$ could also be an extremum. Then you might get an awful lot of candidates if $N$ is a multiple of a high power of $2$: divide your square into dyadic squares and put a diagonal or anti-diagonal in some of them, so that exactly one appears in each row and in each columns. In any case, you should probably start with the continuous case, where computations should be slightly easier (integrals instead of sums) and the arithmetic properties of $N$ will not enter the picture. | |
| Mar 10, 2017 at 12:10 | comment | added | O. Richard | @BenoîtKloeckner No I haven't, but what you said makes sense. In my situation, I am more interested in fixing $\nu$ to be the uniform distribution on $\Xi$. Is there a connection between the case for $\nu$ is uniform over $j=i+N/2$ and $\nu$ is uniform over $\Xi$? | |
| Mar 10, 2017 at 11:48 | comment | added | Benoît Kloeckner | The maximizer might not be unique. Have you considered the case where $\mu$ is the identity coupling (comonotonic distribution in your words), and $\nu$ is uniform over the set $j=i+N/2$ modulus $N$ ($N$ even)? A rough estimate seems to indicate both $\nu$ and the ``countermonotonic'' distribution are at distance $1/2$ (for some, maybe all $p$) of $\mu$. | |
| Mar 10, 2017 at 11:25 | history | edited | O. Richard | CC BY-SA 3.0 |
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| Mar 10, 2017 at 3:31 | history | edited | O. Richard | CC BY-SA 3.0 |
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| Mar 10, 2017 at 3:00 | history | edited | O. Richard |
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| Mar 10, 2017 at 2:54 | history | edited | O. Richard | CC BY-SA 3.0 |
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| Mar 10, 2017 at 2:01 | comment | added | O. Richard | @VictorKleptsyn Yes, you're right. Essentially we want to maximize a convex function over a convex set, thus the maximum would be obtained at an extreme point, and in two dimensional case, this is the permutation matrix. However, since maximizing a convex function is generally hard, we cannot expect a solution for arbitrary distribution $\nu$, but I hope for some special $\nu$ such as the uniform distribution, we can obtain some result. | |
| Mar 10, 2017 at 1:17 | comment | added | Victor Kleptsyn | If I'm not mistaken, permutation measures $\frac{1}{N} \sum_i \delta_{i \sigma(i)}$ are exactly the extremal points of the polytope of the probability measures with uniform marginals. And the maximum distance (to any distribution, in particular, to the uniform one) should be obtained at one of these points. | |
| Mar 9, 2017 at 22:40 | review | First posts | |||
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| Mar 9, 2017 at 22:38 | history | asked | O. Richard | CC BY-SA 3.0 |