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Separability of the Wasserstein space with respect to $W_2(\cdot,.) +|\phi(\cdot) - \phi(.)|$

I would be thankful, if someone could give me some short proof or reference for the following problem. Given a lower semi-continuous and geodesically convex functional $\phi$ on the Wasserstein ...
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25 views

Admissible transference plans, characterizing a condition on measures.

In Villani's, Topics in optimal transportation, pg.18 we find that for a borel measure $\pi\in X\times Y$, where $(X,\mu)$ and $(Y,\nu)$ are probability measure spaces, the condition \begin{equation} ...
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26 views

The dual of transporting problem

So basically I'm trying to figure out what does a certain variable in dual of transporting problem mean. Transporting problem in matrix form: (We are searching for a min cost of transferring goods ...
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1answer
24 views

Continuity of subdifferential mapping

I'm reading Cedric Villani's book topics in optimal transportation, and I have a problem on page 53: If $\varphi$ lower semi-continuous, then the subdifferential mapping $\partial\varphi$ is always ...
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28 views

Definition of “deterministic coupling” [Villani]

I'm currently reading through "Optimal transport, old and new" by Cédric Villani. In the first chapter, he defines a coupling of two probability spaces $(\mathcal{X},\mu)$ and $(\mathcal{Y},\nu)$ as a ...
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21 views

Reference request: Time-optimal trajectories

I am looking for some lecture notes or a textbook for time-optimal trajectories. Any help is greatly appreciated. I am having plenty of trouble with understanding switching $C^+$ and $C^-$ curves.
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52 views

Wasserstein metric: conditions for the existence of minimizer and duality

Let $(X,d)$ be a metric space and let $\mathcal P(X)$ be the set of all Borel probability measures on $(X,d)$. The Wasserstein distance on $\mathcal P(X)$ is given by $$ W_d(\mu,\bar\mu):=\inf_{M\in ...
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45 views

PDE-Based Triangle Inequality for Optimal Transportation

Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and ...
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85 views

Wasserstein distance from a Dirac measure

http://en.wikipedia.org/wiki/Wasserstein_metric I would like to prove that $$W^1(μ,δx_0)=∫d(x_0,y) μ(dy)$$ let $$γ∈Γ(μ,δx_0)$$ Can we say that it is the product of its marginal distributions ...
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339 views

Transportation problem: optimal solution

So I have an issue with finding the optimal solution (the lowest costs) to a transportation problem. Given the following problem, with $A$ the depots, $B$ the destinations and $C$ the $(i,j)$ matrix ...
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41 views

Why is the shift the optimal plan between $[0,1)$ and $[1,2)$ (with distance-squared cost function)?

Example 1.3 of Optimal and Better Transport Plans reads Consider the task to transport points on the real line (equipped with the Lebesgue measure) from the interval [0, 1) to [1, 2) where the ...
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72 views

Spacing nodes by moving the shortest distance possible.

I have a list of N nodes with positions $(x, y)$ each. I want to move each node the shortest possible distance such that every node is placed on the radius $R$ from at least one other node, and is at ...
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115 views

Cyclically monotone sets on four points

A subset of $\mathcal{X} \times \mathcal{Y} \subset \mathbb{R}^d \times \mathbb{R}^d$ is cyclically monotone if $\sum_{i=1}^n \langle x_i,y_i\rangle \ge \sum_{i=1}^n \langle x_i,y_{i+1}\rangle$, where ...