# Questions tagged [optimal-transport]

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82 questions
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### Relative Entropy and the Wasserstein distance

Can anyone give an informative example of two distributions which have a low Wasserstein distance but high relative entropy (or the other way around)? I find the Wasserstein defined (for some $p$) as ...
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### Construction of optimal map

I am not able to show the last step in the proof of the following Theorem: Suppose $X$ and $Y$ are Polish spaces, $\pi$ is a probability measure on $X \times Y$ and $\mu$ is a probability measure on ...
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### Generalized Rockafellar theorem

I am trying to understand the proof of the following Theorem Let $X, Y$ be metric spaces and $c: X \times Y \to \mathbb{R}$ be a continuous and bounded function and let $\Gamma \subset X \times Y$ be ...
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### Two basic questions in Wasserstein spaces

We denote by $P (\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^{d}$ and for $p\geqslant 1$ the Wasserstein space by \begin{equation*} P^p (\mathbb{R}^{d}) = \{ \mu \in P(\mathbb{R}...
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### Wasserstein distance of two flat triangles

Is there a way to compute the Wasserstein distance of two flat triangles that share a commen edge? That is, assume you have two triangles in $\mathbb{R}^2$ that share a commen edge. Further, assume ...
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### Wasserstein attains its infimum

let $(\mathcal{X},d)$ be a Polish space. For $p\geq1$ let $\mathcal{P}_p(\mathcal{X})$ be the space of all Borel probability measures $\mu$ on $\mathcal{X}$ such that \begin{equation} \mathbb{E}_\mu\...
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### Exercise 1.21 of Villani Topics in Optimal Transportation

I'm stuck on this exercise, can you please help me? Since the text is a bit long I attach here below the the screen of the text. Following the notation of the book, $\Pi(\mu,\nu)$ is the set of ...
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### Realize a coupling in the target space via a measure on the source space

Consider two product measurable spaces $\left(X \times Y,\mathcal{X} \otimes \mathcal{Y}\right)$, $\left(X' \times Y',\mathcal{X'} \otimes \mathcal{Y'}\right)$ with the usual product sigma-algebra, ...
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### Coupling via push-forward from a source space

Consider a measurable function $g$ mapping a probability space $\left(\Omega,\mathcal{F},\mu\right)$ to a product measurable space $\left(T,\mathcal{T}\right)$ with cartesian product $T = X \times Y$ ...
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### Completeness of Wasserstein space

In this MO question it is said that: On a complete, non necessarily separable metric space $E$, the set $P_r(E)$ of all Radon probability measures with the Wasserstein-Kantorovich metric $W_d$ is ...
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### Proof that the optimal transport cost is lower semicontinuous

In Chapter 6 of Villani's Optimal Transport: Old and New, it is stated that ... the Wasserstein distance is lower semicontinuous on $P(\mathcal{X})$ (just like the optimal transport cost $C$, for ...
The transportation problem is a well-studied problem in operations research. Given sources $i\in\{1, \ldots, n\}$ and destinations $j\in\{1, \ldots, m\}$, we seek to minimize the total cost of ...
I am reading Villani's Optimal transport: Old and new. Theorem 4.1 concerns the existence of an optimal coupling between any two Polish probability spaces $(\mathcal{X}, \mu)$ and $(\mathcal{Y}, \nu)$...