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.

# Questions tagged [optimal-transport]

300 questions

**23**

votes

**2**answers

### Closed-form analytical solutions to Optimal Transport/Wasserstein distance

Kuang and Tabak (2017) mentions that:
"closed-form solutions of the multidimensional optimal transport problems are relatively rare, a number of numerical algorithms have been proposed."
I'm wondering if there are some resources (lecture notes,…

random_shape

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votes

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### 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 $$γ=μ×δx_0 $$ and then apply Fubini's theorem? But…

user149901

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votes

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### 2-Wasserstein distance between empirical distributions

I am trying to validate the earth-mover distance implementation from the python optimal transport library https://pot.readthedocs.io/en/stable/all.html#module-ot. It would be great if somebody could point at any flaws in what I write below.
Given…

Marc

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**6**

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**1**answer

### The Lebesgue measure as energy minimizer?

Assume that $K:\mathbb{R}_{\geq0} \to\mathbb{R}_{\geq0}$ is a decreasing function. If $\nu$ is a probability measure on the unit circle, define its energy
$$
\mathcal{E}(\nu):=\int K(|x-y|)d\nu(x)d\nu(y).
$$
Is there a simple argument to show that…

Kostya_I

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**6**

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### Name for a cardinality-constrained transportation problem variant

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 shipping materials from sources to destinations. Each…

josliber

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### Is the expectation value Lipschitz for the Wasserstein metric?

Consider the space $M$ of Borel measures on the real unit interval $[0,1]$, equipped with the 1-Wasserstein metric $d_W$ (or "Earth mover's distance").
The expected value is then a map $M\to [0,1]$ given by:
$$
p \mapsto E_p := \int_{[0,1]} x \,…

geodude

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**5**

votes

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### Understanding the velocity field in the Benamou–Brenier formulation

Disclaimer: theoretical physicists here! I apologize in advance for some sloppy terminology.
I recently found an interest in optimal transport theory for some potential application to my research field in physics and I have been trying to study some…

ggg98

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### How is Optimal Transport algorithmically related to the Assignment Problem?

In optimal transport, we calculate the distance between two probability measures $\mu$ and $\nu$ over the compact set $[a,b]\subset\mathbb R$, using the Earth Movers distance which is a special case of the Wasserstein…

develarist

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### Alternate definition of Entropy

I am going through the book Computational Optimal Transport by Peyré and Cuturi.
In it (see Formula (4.1), P. 65/209) I came across the definition of the discrete entropy, which is not what I expected:
$$
H(P) = -…

dba

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**5**

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### What is the fastest route to drop off weight when time is proportional to weight x distance?

You have a lorry at the starting point which is carrying all the parcels for the day.
$$\rm Time\ taken = Total\ Lorry\ Weight \times Distance\ travelled $$
After visiting each zone you have to return to the starting point.
Lorry's weight is $30$.…

Ben Franks

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### Characterization of Wasserstein convergence

Let $(X,d)$ be a complete metric space and define
$$\mathcal{P}_2(X) := \{ \mu \text{ Borel probability measure} \mid \int_X d^2(x,x_0) d\mu(x) < \infty \text{ for some } x_0 \in X \}$$
endowed with the Wasserstein distance
$$ W^2_2(\mu, \nu) =…

Bremen000

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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 any lower semicontinuous cost function $c$; recall…

12qu

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### Derivation of the weak transport equation

Let $\rho$ be a density on some space $M$. For all practical purposes some subspace of $\mathbb R^n$. Let $v$ be a smooth vector field with flow $\Phi_t$. The transport equation, as far as I understand, governs the time evolution of $\rho_t$, which…

Marlo

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### 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 cost of moving one point to another is the squared…

Quinn Culver

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### Maximiser of $W_1(\mu, \nu)$ can be changed outside of $\text{conv}(\text{supp}(\mu) \cap \text{supp}(\nu))$ (under additional assumptions)

Let $(X, \| \cdot \|)$ be a reflexive Banach space and $\mathbb{P}_n$, $\mathbb{P}_r$ be measures on $X$.
Let the support of $\mathbb{P}_r$, $M := \text{supp}(\mathbb{P}_r)$ be a weakly compact set and $$P_M \colon D \to M, \qquad x \mapsto…

Ramanujan

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