All Questions
Tagged with wasserstein probability-theory
16
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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. ...
2
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86
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What is a wasserstein barycenter?
I am currently studying a paper on Wasserstein Fair Classification. Several places they mention the Wasserstein barycenter, weighted barycenter distribution or the Wasserstein barycenter distribution. ...
- 21
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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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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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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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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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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{...
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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
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Wasserstein distance between a distribution of a random variable and the distribution of its projection onto a subset of its sample space
Consider a random variable $x$ with a distribution $p_x$ supported on whole of $\mathbb{R}^n$ ($n$ being a natural number). Let $S \subset \mathbb{R}^n$. Let $y = {\rm proj}_S(x)$ denote the ...
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337
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An inequality about the 2-Wasserstein distance
Let $W_2(\mu,\nu)$ denote the $2$-Wasserstein distance between two given probability measures $\mu$ and $\nu$ on $\mathbb R^n$. For a probability measure $\mu$ and $f:\mathbb R^n\to \mathbb R^n$, let $...
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367
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Can we bound the L1 distance between densities by Wasserstein distance of measures
Let $\mu_1$ and $\mu_2$ be two probability measures over a closed interval $[a, b]$, with respective density functions $\phi_1$ and $\phi_2$. Is there a way to bound the $L^1$ distance of the ...
- 317
3
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222
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Absolutely continuous curves in Wasserstein distance and measurability.
Let $(X, d, \mu)$ be a metric measure space. Let $P^1(X)$ denote the space of probability measures on $(X,d)$, which have finite first moments, that is:
\begin{equation}
\nu \in P^1(X) \implies \int d(...
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Concentration inequalities for random measures
For random variables $X_1,\dots,X_n$ with common mean $\mathbb{E}[X_i]=\mu$ and common bounds $a\leq X_i\leq b$, we have the very useful Hoeffding's inequality:
$$\mathbb{P}\left(\left|\mu -\frac1n\...
- 1,261
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Wasserstein distance of convolution of measures
Let $\mu_1, \mu_2,\nu_1,\nu_2$ be measures on $\mathbb{R}^d$. It is well known that the $p$-Wasserstein distance satisfies
$$\mathcal{W}_p(\nu_1 *\mu_1, \nu_1 *\mu_2) = \mathcal{W}_p(\mu_1,\mu_2),$$
...
- 1,261
1
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Extension of Kantorovich-Rubinstein inequality.
Let $(\mathcal{X}, \Sigma)$ be a Polish metric space, endowed with the Borel $\sigma$-algebra. Let $\mathscr P$ be the space of probability measures on $\mathcal X$ and $\mathscr P^1$ be defined as
$$\...
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