Questions tagged [wasserstein]
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Can a finite Wasserstein metric on Euclidean support be embedded in a Euclidean space?
Thanks for everyone's help with understanding finite metric embeddings in Euclidean space. I have a follow-up question.
Say we have the Wasserstein distance between $n$ distributions in Euclidean ...
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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. ...
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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 ...
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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-...
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Wasserstein distance inequality
Suppose $(\Omega, \mathcal F, \mathbb P)$ is a probability space. Suppose $X, X', Y, Y'$ are random variables.
Denote $W_1$ the Wasserstein-1 distance between $\mathbb P_X$ and $\mathbb P_{X'}$
and $...
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Bound for expected value under Wasserstein metric
I'm reading a paper and the following result is presented:
$$ (\mathbb{E}_{F}[\|\mathbf{X}\|^k])^{1/k} \leq (\mathbb{E}_{F_{0}}[\|\mathbf{X}\|^k])^{1/k} + \epsilon, \ \forall F\in\mathcal{B}_{p}(F_{0},...
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Regularity of Kantorovich potentials for general cost function
I know De Philippis and Figalli have a paper studying the regularity of the Kantorovich potential. In Theorem 3.3, the authors show that the potential is $C^{k+2, \beta}$ if the density functions of ...
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proof that the wasserstein space is no manifold
This is my first question on this platform, I appreciate any suggestions on how to improve my question.
why is the Wasserstein space no manifold and in which way is its structure somehow similar to a ...
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Understanding different norms in the p-Wasserstein distance
The generalized p-Wasserstein distance, for $p\geq 1$, is given by
$$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 $\xi_1$...
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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. ...
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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 ...
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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 ...
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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:...
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Is there a meaning of distances for 0<p<1 for Wasserstein distance?
The wikipedia link for Wasserstein metric is defined for $p\in[1,\infty)$.
https://en.wikipedia.org/wiki/Wasserstein_metric Given some data the distance can be calculated using an optimization ...
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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 ...
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