All Questions
Tagged with wasserstein optimal-transport
29
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There is a unique coupling between a probability distribution $\mu$ and a degenerate distribution $\mu_0$. [duplicate]
By degenerate distribution I intend https://en.wikipedia.org/wiki/Degenerate_distribution.
I cannot see why the set of couplings between some probability measure and a constant would be a singleton. ...
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22
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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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1
answer
24
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Bound for expected value under Wasserstein metric
I'm reading a paper and the following result is presented:
$$ (\mathbb{E}_{F}[\|\mathbf{X}\|^k])^{1/k} \leq (\mathbb{E}_{F_{0}}[\|\mathbf{X}\|^k])^{1/k} + \epsilon, \ \forall F\in\mathcal{B}_{p}(F_{0},...
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Regularity of Kantorovich potentials for general cost function
I know De Philippis and Figalli have a paper studying the regularity of the Kantorovich potential. In Theorem 3.3, the authors show that the potential is $C^{k+2, \beta}$ if the density functions of ...
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Showing that a minimzing sequence for the Wasserstein Variance functional must be tight
I am trying to understand this 2011 paper by Agueh & Carlier [https://www.ceremade.dauphine.fr/~carlier/AC_bary_Aug11_10.pdf] where they introduce the notion of barycenter in the 2-Wasserstein ...
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What is the image of a transference plan?
Let $\mu, \nu$ be two positive Borel measures on $\mathbb{R}^d$ with the same mass. A probability measure $\pi$ on $\mathbb{R}^d\times\mathbb{R}^d$ is called a transference plan from $\mu$ to $\nu$ if:...
2
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55
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Bregman divergence from Wasserstein distance
I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance.
More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...
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2
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156
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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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1
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196
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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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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 ...
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47
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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 $...
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171
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How to derive the $W_1$ Wasserstein distance written with the quantile functions (i.e. with inverse cumulative distribution functions) [closed]
INTRODUCTION. Either downloading the slides on Optimal Transport (OT) from the Marco Cuturi website (go to section "Teaching ENSAE --> OT --> Optimal Transport (Spring 2023)
[slides]" ...
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Optimal Mass Distribution Minimizing Average 2-Wasserstein Distance to a Set of Mass Distributions
Given a fixed set of $n$ points in 2D (Earth Movers distance Prpblem), $P = \{p_1, p_2, ..., p_n\}$, I am trying to find the mass distribution $\bar{M}$ that minimizes the average 2-Wasserstein ...
2
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244
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Derive the $d=1$ form of the Wasserstein distance
QUESTION
Given
How to pass from the general definition of the Wasserstein distance (let's call it Equation (1)):
to the closed forms with d=1, here below (let's call them Equation (2) and Equation(3)...
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Boundedness of $\dfrac{W_2(\mu_1+\varepsilon (\mu_2-\mu_1),\mu_1)}{\varepsilon}$ for 2 -Wasserstein metric
Let $\mathcal{P}_2(\mathbb{R}^{n})$ the space of Borel probability measures of finite second moment in $\mathbb{R}^{n}$ equipped with the $2$-Wasserstein metric $W_2$. Let $\mu_1$, $\mu_2 \in \mathcal{...