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Mathematics Stack Exchange

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.

388 questions
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A proof of Kantorovich duality

Disclaimer This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $X$ and $Y$ be Polish spaces. Let $P(X), P(Y)$ be the spaces of all Borel probability…
Akira
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Optimal transport for point sets

I have been examining the literature on optimal transport, and typically it is introduced as an arbitrary number of probability measures $\mu_0, \mu_1, \cdots$ on an n-dimensional manifold $N$. The situation I wish to examine is a lot simpler -…
Daniel Hogg
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Not seeing symmetry of Rubenstein-Kantorovich dual form of the Wassterstein-1 distance

Let's just jump right in and define the dual form of the Wasserstein-1 distance for 2 measures: $\mu$ and $\nu$ as follows: $$W_1(\mu, \nu ) = \sup_{f \in \text{1-Lipschitz} } \int f d\mu - \int f d\nu$$ The metric implies the following inequality,…
jeffery_the_wind
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Fundamental theorem of optimal transport

I am trying to read Ambrosio's user guide to optimal transport and I am having some difficulties with the following Theorem Theorem 1.13 (Fundamental theorem of optimal transport) : Assume that $c : X × Y → R$ is continuous and bounded from below…
roi_saumon
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Optimal transport - Earth mover's distance

I want to calculate the distance between two probability measures $\mu$ and $\nu$ over the compact set $[a,b]\subset\mathbb R$, using the Wasserstein formula: $$W=\inf_\varphi\int|x-y|\,{\rm d}\varphi(x,y),$$ where $\varphi$ is interpreted as a…
Coralio
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Computing 1- or 2- Wasserstein distance between collections of point masses

Is either $W_1$ or $W_2$ (the Wasserstein distances) available in closed form when comparing two collections of point masses? To be specific, let $p = \sum_{i=1}^n \delta_{x_i}$ and $q = \sum_{j=1}^m \delta_{y_j}$ for two arbitrary sets of points…
p-value
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Similarity between two optimal transport plans

I have recently been studying the theory of computational optimal transport and am very interested in how to describe the similarity between two optimal transport plans. Specifically, suppose $P_1,Q_1,P_2,Q_2$ are probability measures on $\Omega…
Mzxr
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$W_p (\mu_m, \mu) \to 0$ if and only if $\mu_m \overset{\ast}{\rightharpoonup} \mu$ and $\int_X |x|^p \mathrm d \mu_m \to \int_X |x|^p \mathrm d \mu$

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. Let $X := \mathbb R^d$, $p \in [1, +\infty)$, and $$ \mathcal P_p(X) := \left \{\mu \in \mathcal P(X)…
Akira
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Name for Monge-Kantorovich transportation problem variant with unequal total mass

I'm interested in a variant of the transportation problem and cannot find a reference for the problem I'm thinking of. In the original Monge-Kantorovich problem about continuous transport, the total mass of $m$ objects are to be moved to the same…
Greenteamaniac
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Questions on the book of Ambrosio-Gigli-Savaré <>

I am learning the book of Ambrosio-Gigli-Savaré "Gradient flow". However, I am blocked at the proof of proposition 8.5.2 (Optimal displacement maps are tangent). I don't understand where $\nabla \phi_\epsilon$ come from and why absolute continuity…
YJ_
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the relations between the pushfoward of a random vector and the pushfoward of the linear projections of the random vector

Let $([0,1]^n, \mathcal{B}, P)$ be a probability space where $\mathcal{B}$ is the Borel $\sigma$-algebra and $P$ is a probability measure. Let $f:[0,1]^n\to \mathbb R^n$ be a measurable function (a random vector). The pushfoward measure $Q=f_{\#}…
sam
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Regularity of Kantorovich potentials

I'm looking for a reference of a result in Optimal Transport. I know that if $\Omega$ is a compact subset of $\mathbb R^n$ and $c\in C^1$ (the transportation cost), then $c$ is Lipschitz continuous on $\Omega\times\Omega$ and hence all the…
Jeji
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I have a question about wasserstein distance

Given the wasserstein distance of order two, $$d(\mu_{1},\mu_{2})^{2}=inf_{p\in P(\mu_{1},\mu_{2})}\int_{R^{n}\times R^{n}}|x-y|^{2}p(dxdy)$$ Is there $d(\mu_{1},\mu_{2})=d(\mu_{2},\mu_{1})$ or $d(\mu_{1},\mu_{2})=-d(\mu_{2},\mu_{1})$?
Sha
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Construct the $c$-transform $(\overline \varphi, \overline \psi)$ of $(\varphi, \psi)$

Disclaimer This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $X,Y$ be Polish spaces and $c:X \times Y \to \mathbb [0, +\infty)$ lower…
Akira
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Composition of two transport maps and pushforward operator

Given two absolutely continuous probability measures $\mu,\sigma \in \mathcal P_2(\mathbb R^n)$ and two maps $T_1, T_2$ such that $$(T_1 \circ T_2)_\#\sigma =\mu$$ where $(\cdot)_{\#}$ denotes the pushforward operator. I saw that it is a general…
Tesla
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