All Questions
Tagged with wasserstein probability
14
questions
1
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Connect definition of 1st-order Wasserstein distance given CDFs and general definition of Wasserstein distance
In this post, the definition of the 1st-order Wasserstein distance is
$\int_{-\infty}^{\infty} |F_1 (x)-F_2(x)| dx$
In Wikipedia, I see something completely different.
How do I connect the 2 ...
- 107
0
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0
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18
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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 ...
- 11
3
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1
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477
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Quantitative bound on Wasserstein distances by $L^p$ distances?
Given two smooth probability densities $f$ and $g$ on $\mathbb{R}$ (or $\mathbb{R}_+$) with finite $p$-th moments. I am wondering if anyone is aware of some explicit upper bound on $W_p(f,g)$ in terms ...
- 2,860
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37
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Sufficient Conditions on Metric Space for Wasserstein Distance?
For a metric space $M$ and the Wasserstein 1-distance $W_1$, what qualifying assumptions do we need for $M$? I have seen we only need $M$ be compact, $M$ be a Polish space, or $M$ be a Radon space. ...
- 71
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85
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How to intuitively view Wasserstein distance dual as moving earth
The Wasserstein-1 distance can be viewed as the minimum amount of work needed to move one distribution to another distribution, as if the distributions were like piles of earth. The typical definition ...
- 195
1
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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]" ...
- 349
3
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1
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299
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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{...
- 395
3
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1
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186
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Relations between Kolmogorov-Smirnov distance and Wasserstain distance
Let's have $X,Y \in \mathbb{R}$ with probability measures $\mu, \nu$, then the Kolmogorov-Smirnov distance is defined as follows
$$ d_K(X,Y)=\underset{x \in \mathbb{R}}{sup}\{|F_X(x) - F_Y(x)|\} $$
...
- 155
2
votes
1
answer
810
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Relation between Wasserstein distance and distribution convergence
Let's have a succession $X_n$ of real value random variable and another real value random variable X, then
$$ X_n \overset{d}{\xrightarrow{}} X \iff \lim_{n \to \infty}{}d_K(X_n,X) = 0 $$
where $d_K(X,...
- 155
1
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1
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61
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Is the space of probability measure dense in the space of probability measure with density?
Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of all probability measures with finite second moment. Let $\mathcal{P}_2^*(\mathbb{R}^n)$ be the subset of $\mathcal{P}_2(\mathbb{R}^n)$ such that any ...
- 395
1
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1
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176
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What is the p-th moment finite in the definition of Wasserstein space?
I am confused about the following notation:
For a simple case, let $X=R^d$ or $X=R$. What dose
$$
\int_X \|x\|^pd\mu(x)
$$
mean for a Borel probability measure $\mu$?
For $X=R$, then $x\in R$ is a 1-...
- 1,520
0
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0
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22
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Is there any relationship between the following two expectations?
Is there any relationship between the following two expectations?
$\mathbb{E}_{\mathbb{Q}}[\|\boldsymbol{\tilde{\xi}} - \boldsymbol{\tilde{\xi}}^{\prime}\|]$, and
$\mathbb{E}_{\mathbb{Q}}[\|\mathbf{A}...
- 41
1
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0
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105
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2-Wasserstein barycenter of uniform distribution on ellipsoid
Let $A$ be a positive-definite symmetric matrix. Consider the ellipsoid $E = \{ x \in \mathbb{R}^n \colon <x A^{-1} x> \leq 1 \}$.
Now consider uniform distribution $\mu_1, \ldots, \mu_n$ on $...
- 87
1
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70
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Density associated with Wasserstein geodesic
Suppose we have two absolutely continuous distribution functions $F$ and $G$ with densities $f$ and $g,$ respectively. Assuming a quadratic cost function, the Wasserstein geodesic at time $t$ is the ...
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