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Results tagged with wasserstein-distance
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user 33741
9
votes
Proving the inequality involving Hausdorff distance and Wasserstein infinity distance
EDIT: answer 2 below is completely false, as pointed out by the OP. However this is such a typical example of wishful thinking that I believe it is worth leaving for the posterity. (I'll record it bel …
- 5,380
6
votes
Accepted
Perturbation of Wasserstein distance: looking for references
You can find this in Villani's "small book", Theorem 8.13 in [Villani, C. (2003). Topics in optimal transportation (Vol. 58). American Mathematical Soc.]
I can also recommend looking at Filippo Santan …
- 5,380
5
votes
Accepted
Gradient of Wasserstein distance in the sense of Otto's calculus
Yes this is true, formally this follows by the envelope theorem. In an abstract and very smooth setting, the envelope theorem says that for an objective functional depending on a parameter $t$
$$
F(t) …
- 5,380
2
votes
Accepted
Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
$\newcommand{\R}{\mathbb R}\newcommand{\H}{\mathcal H}$Building up on previous comments, and slighlty elaborating. The answer to your question, as is, is NO. Recall that convergence in the Wasserstein …
- 5,380
1
vote
Accepted
Equivalent definition of the Kantorovich-Fisher-Rao distance
Well, when we wrote the paper we were not really concerned with full rigor at this stage, all we wanted to emphasize was that the "KFR" distance (by now rather the WFR or HK distance, as in Wasserstei …
- 5,380
1
vote
Accepted
Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with ...
$\newcommand{\R}{\mathbb R}$This is only a partial answer in the sense that it will provide $\frac 1p$-Hölder maps, not Lipschitz. I still hope it can help.
Fix a big radius $R>0$ (I like $R\to\infty …
- 5,380
1
vote
Accepted
Gradient flows: evolution of geodesics
As currently asked the answer is NO, because your desired upper bound already fails for $t=0$ (or equivalently, $t=1$). Indeed, it is well understood that the small-time deviation along the heat flow, …
- 5,380