# 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 deal with ill conditioned optimization problem (Sinkhorn barycenter)

Problematic (Debiased Sinkhorn barycenter, proposed by H.Janti et al.): Let $\alpha_1, \ldots, \alpha_K \in \Delta_n$ and $\mathbf{K}=e^{-\frac{\mathrm{C}}{\varepsilon}}$. Let $\pi$ denote a sequence ...
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### 1D-Wasserstein-Barycenter closed form

I just want to verify that my reasoning here is correct. It feels very basic but I can't seem to find this result in any textbook. It is well-known that the 1-Wasserstein distance in 1D can be ...
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### On the Optimal Transport Distance in Equivalent Martingale Measures

I've been working on this idea for a while and could use some guidance. Right now I'm wondering if the formulas I am writing out make sense. Any help is very much appreciated. Problem Statement Idea ...
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### Optimal transport - Monge plan - measure theoretic calculation

In my Optimal transport course we dealt with the Monge optimal transportation problem. We have: some open sets $X, Y \subset \mathbb{R}^d$, a cost function $c : X \times Y \rightarrow [0, +\infty ]$....
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### Definition of marginal probability measures using test function spaces

This is a question from Villani's Topics in Optimal Transportation. Let's define $\mu, \upsilon$ probability measures on some measure spaces $X, Y$ respectively, and $\pi$ a probability measure on the ...
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### Example of finding the lower semi-continuous convex function whose subdifferential contains the support of an optimal transport

Related to this post, but slightly altered. Let $\mu$ uniform on $\{0\}\times [0,1]$, $\nu$ uniform on $\{-1,1\}\times [0,1]$. Consider the Kantorovich optimal transport problem of $\mu$ to $\nu$ with ...
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### Simplifying large scale nonlinear numerical optimization problem

I want to solve the optimization problem below numerically using GAMS. Given the formulation, I was wondering if there are suitable ways of transforming and or approximating the given function to ...
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### Distance between empirical distribution and population

Suppose we have $N$ points $z_1, \dots, z_N$. From this we can form the empirical distribution $\mu_z = \frac{1}{N} \sum_{i=1}^N \delta_{z_i}$. Now suppose we sample $N$ i.i.d. points from $\mu_z$, i....
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### Density of the maximum of 2 random variables with arbitrary dependence

Let $(X,Y)$ be a random vector taking values in $\mathbb{R}^2$ such that both $X$ and $Y$ are (marginally) distributed uniformly over the interval [0,1]. Let $Z= \max(X,Y)$ and $f$ denote the density ...
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### Wasserstein Metric Inequality

This is the exercise: This exercise shows that “spreading out” probability measures makes them closer together. Define the convolution of a measure by: for any probability density function $\phi$, let ...
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### If $f > 0$, $\ker(L)$ contains only constant functions, where $L = - \Delta + \nabla (- \log(f)) \cdot \nabla$ (Villani, Subsec. 7.6)

In subsection(s) 7.5 (and 4.1) of Topics in Optimal Transportation, Cedric Villani states the following (I paraphrase): Take a probability measure $\mu \in \mathcal P_2(\mathbb R^n)$ with finite ...
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### minimizing Earth Mover Distance

So I have a discretized magnitude spectrum $S \in \mathbb{R}^n$ ($n$ number of bins), and a set of frequencies $f_1, f_2, ..., f_m$ (not necessarily corresponding to any of the discretized bin ...
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### Are linear interpolation curves on Wasserstein spaces absolutely continuous?

Let $\mathcal{P}_2$ the space of absolutely continuous probability measures on $\mathbb{R}^d$ with finite second moment equipped with the $2$-Wasserstein metric. Fix $\mu_0, \mu_1 \in \mathcal{P}_2.$ ...
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### If $z=(x, v) \sim \pi(dx)\eta(dv)$, find $\eta(dv)$ such that $v'\sim\varpi$ where $z'=\psi(z)$ and $\psi$ invertible

I have a variable $z = (x, v)$ distributed according to $\lambda(dz) = \pi(dx)\eta(dv)$, where $\pi$ is fixed and known, but $\eta$ is unknown. I also have an invertible transformation $\psi$ and I ...
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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 $\... • 13 0 votes 0 answers 31 views ### Equivalent condition on marginals Let$\mu$be a probability measure on$X$(compact),$v$a probability measure on$Y$(compact), and let$\gamma$be a probability measure on$X \times Y$: if $$\forall \psi \in C(X), \int_X \psi(x)d\... • 45 2 votes 0 answers 130 views ### Distance between a distribution and an empirical distribution of samples Looking for a reference on what I think is a standard result. For a measure \mu \in P(\mathbb{R}^n), let y_1, \dots, y_N \sim \mu be independent samples. The empirical distribution associated with ... • 2,039 0 votes 0 answers 21 views ### how to solve for wasserstein duality easily in a special case when 2-Wasserstein inequality constraint is binding I was going through this nice paper ” A Simple and General Duality Proof for Wasserstein Distributionally Robust Optimization”, and one quick qu on applying Theorem 1 to my poject: What if in my ... • 61 1 vote 1 answer 95 views ### Optimal Transport of radial measures This is the exercise i). Find a transport map between d\mu_d=\frac{1}{\pi}\mathbb{I}_{B_1(0)}dxdy and d\mu_c=\frac{1}{8\pi}\left(4-|x|^2\right)_+dxdy. ii) Compute C^K_{\rho}(\mu_d,\mu_c) where ... 0 votes 0 answers 108 views ### Solve y=\operatorname{erf}(x+c)+\operatorname{erf}(x-c) for x \newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}} Is there a closed form solution for y=\erf(x+c)+\erf(x-c)? More specifically, I want to solve$$\erf(\frac{y}{\sqrt{2}})... • 11.8k 0 votes 0 answers 47 views ### What is the conventional definition of$d(x,y)^{p}$? In Optimal transportation, and more precisely in "Optimal Transport: Old and New" (Definition. 6.1, on page 106 - actually on page 111 out of 998, in this link), the Wasserstein distance$...
I'm trying solve the following problem: Find a sequence $T_n$ of transport maps from $\mathbb{I}_{[0,1]}dx$ to $\mathbb{I}_{[0,1]}dx$ such that $T_n$ converges weakly to $\frac{1}{2}$. Deduce that the ...