Questions tagged [wasserstein]
The wasserstein tag has no usage guidance.
66
questions
0
votes
0
answers
25
views
There is a unique coupling between a probability distribution $\mu$ and a degenerate distribution $\mu_0$. [duplicate]
By degenerate distribution I intend https://en.wikipedia.org/wiki/Degenerate_distribution.
I cannot see why the set of couplings between some probability measure and a constant would be a singleton. ...
0
votes
0
answers
18
views
1D-Wasserstein-Barycenter closed form
I just want to verify that my reasoning here is correct. It feels very basic but I can't seem to find this result in any textbook. It is well-known that the 1-Wasserstein distance in 1D can be ...
- 11
0
votes
0
answers
24
views
Does this relation between Wasserstein distances hold: $W_1(\mu,\nu)\leq W_2(\mu,\nu) \leq ... \leq W_\infty(\mu,\nu)$?
I stumbled upon this interesting statement in this paper:
"One interesting observation is that the Wasserstein ambiguity
set with the Wasserstein order p = 2 is less conservative, because the 2-...
- 99
1
vote
1
answer
23
views
Bound for expected value under Wasserstein metric
I'm reading a paper and the following result is presented:
$$ (\mathbb{E}_{F}[\|\mathbf{X}\|^k])^{1/k} \leq (\mathbb{E}_{F_{0}}[\|\mathbf{X}\|^k])^{1/k} + \epsilon, \ \forall F\in\mathcal{B}_{p}(F_{0},...
- 286
0
votes
0
answers
21
views
Regularity of Kantorovich potentials for general cost function
I know De Philippis and Figalli have a paper studying the regularity of the Kantorovich potential. In Theorem 3.3, the authors show that the potential is $C^{k+2, \beta}$ if the density functions of ...
- 11
1
vote
0
answers
34
views
proof that the wasserstein space is no manifold
This is my first question on this platform, I appreciate any suggestions on how to improve my question.
why is the Wasserstein space no manifold and in which way is its structure somehow similar to a ...
- 11
0
votes
0
answers
31
views
Understanding different norms in the p-Wasserstein distance
The generalized p-Wasserstein distance, for $p\geq 1$, is given by
$$d_W(Q_1,Q_2):=inf \left\{\int_{\Xi_2}||\xi_1-\xi_2||^p \Pi(d\xi_1,d\xi_2)\right\}$$
where $\Pi$ is the joint distribution of $\xi_1$...
- 99
1
vote
1
answer
46
views
Connect definition of 1st-order Wasserstein distance given CDFs and general definition of Wasserstein distance
In this post, the definition of the 1st-order Wasserstein distance is
$\int_{-\infty}^{\infty} |F_1 (x)-F_2(x)| dx$
In Wikipedia, I see something completely different.
How do I connect the 2 ...
- 107
0
votes
0
answers
18
views
Showing that a minimzing sequence for the Wasserstein Variance functional must be tight
I am trying to understand this 2011 paper by Agueh & Carlier [https://www.ceremade.dauphine.fr/~carlier/AC_bary_Aug11_10.pdf] where they introduce the notion of barycenter in the 2-Wasserstein ...
- 11
1
vote
0
answers
16
views
What is the image of a transference plan?
Let $\mu, \nu$ be two positive Borel measures on $\mathbb{R}^d$ with the same mass. A probability measure $\pi$ on $\mathbb{R}^d\times\mathbb{R}^d$ is called a transference plan from $\mu$ to $\nu$ if:...
- 11
3
votes
1
answer
461
views
Quantitative bound on Wasserstein distances by $L^p$ distances?
Given two smooth probability densities $f$ and $g$ on $\mathbb{R}$ (or $\mathbb{R}_+$) with finite $p$-th moments. I am wondering if anyone is aware of some explicit upper bound on $W_p(f,g)$ in terms ...
- 2,860
2
votes
1
answer
83
views
What is a wasserstein barycenter?
I am currently studying a paper on Wasserstein Fair Classification. Several places they mention the Wasserstein barycenter, weighted barycenter distribution or the Wasserstein barycenter distribution. ...
- 21
2
votes
0
answers
55
views
Bregman divergence from Wasserstein distance
I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance.
More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...
- 13.3k
0
votes
0
answers
31
views
Functional E convex and lower semicontinuous implies weakly lower semicontinuous in Wasserstein space
I have a certain functional $E : W_2 \rightarrow \mathbb{R}$, where $W_2$ is the 2-Wasserstein space (metric and separable).
Such functional is convex.
Now, can I state that if $E$ is strongly lower ...
- 21
0
votes
0
answers
37
views
Sufficient Conditions on Metric Space for Wasserstein Distance?
For a metric space $M$ and the Wasserstein 1-distance $W_1$, what qualifying assumptions do we need for $M$? I have seen we only need $M$ be compact, $M$ be a Polish space, or $M$ be a Radon space. ...
- 71