# Questions tagged [optimal-transport]

For questions related to the theory, applications, and computational aspects of optimal transport and related topics such as the Wasserstein (and other transportation cost) distances, the Monge-Ampere equation, metric gradient flows, martingale optimal transport, and optimal matching.

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### How to Correctly Write the Wasserstein (Kantorovich) Distance between Transition Probability Densities of two Markov Processes

I am exploring an idea which is based on computing the Wasserstein (Kantorovich) distance between the empirical transition probability densities of two Itô diffusion processes. Following from this ...
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### Can certain Convex Optimization Problems be interpreted as Optimal Transport Problems?

The theory of optimal transport considers the problem of transporting utilities distributed acoording to a probability measure $\mu$ on $X$ to "targets" distributed according to a ...
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### Kantorovich problem for compact spaces and continuous cost function

I want to prove the existence of a solution to a special case of the Kantorovich problem: Let $(X,\mu)$ and $(Y,\nu)$ be compact probability spaces, i.e. $X$ and $Y$ are compact metric spaces and $\mu$...
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### What is the difference between discrete optimal transport and linear programming?

How unique minima are guaranteed to exist in optimal transport. Why such a thing is not possible in general linear programming problems. What theorem guarantees the existence of unique minima to exist ...
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### Convergence of $\mathbb{E}|\int X(\omega,x) (m_k(dx)-m(dx))|^2$ when $m_k(x)dx$ tends to $mdx$ in Wasserstein 2 distance

In $\mathbb{R}^n$, we suppose that $m_k$, $m$ are probability densities with finite second moment such that $m_k(x)dx$ tends to $mdx$ in Wasserstein 2 distance as $k \to \infty$. If $X(\omega,x)$ is a ...
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### Class of transport maps is not closed in space of all measurable maps?

I am trying to understand the following: Definition: Map $t:X \to Y$ is called a transport map if it is measurable and $t_{\#}\mu=\nu$ where $\mu$ is a measure on $X$ and $\nu$ is a measure on $Y$. ...
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### Does the reverse triangle inequality holds for Wasserstein-1 distance?

Let $(X,d)$ be a separable metric space with associated Borel $\sigma$-algebra $\mathcal{B}(X)$ and the set of Borel probability measures $\mathcal{P}(X)$. For $\mu,\mu'\in\mathcal{P}(X)$ Wasserstein-...
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### Optimal Transport between two Gaussians

Consider the optimal transport map $T$ between $N(\mu_0,\Sigma_0)$ and $N(\mu_1,\Sigma_1)$. I believed that the optimal transport was given by:  T(x) = \mu_1 + \Sigma_1^{1/2} \Sigma_0^{-1/2}(x-\mu_0)...
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### Optimal Transport map between two uniform distribution

Let $\mu$ be a uniform distribution over the interval $[0,1]$, $\nu$ be a uniform distribution over the set $[-1/2,0] \cup [3/2,2]$. What is the optimal transport map from $\mu$ to $\nu$ with ...
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### Derivative of sequence of measures

Let $\mathcal{P}_2(\mathbb{R})$ be the space of probability measures with finite second moment on $\mathbb{R}$, and $(\mu_t)_{t\geq 0}\subseteq \mathcal{P}_2(\mathbb{R})^{\mathbb{R}^{\geq0 }}$ be a ...
### When does there exist a map $T$ such that $(X,Y)$ and $(X,T(X))$ have the same distribution?
Let $\mathcal X\subseteq \mathbb R^d$, $\mathcal Y\subseteq \mathbb R$ be equipped with their Borel sigma-algebras, and $X$ and $Y$ be two respectively $\mathcal X$ and $\mathcal Y$ valued random ...
### Optimal transport map in $\mathbb{R}^n$ with any norm
In the last page of Villani's book "Optimal Transport, Old and New" there's written this: let $N=\Vert\cdot\Vert$ be a uniformly and smooth convex norm on $\mathbb{R}^n$ (i.e. \$\exists\...