All Questions
Tagged with wasserstein probability-distributions
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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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Does this relation between Wasserstein distances hold: $W_1(\mu,\nu)\leq W_2(\mu,\nu) \leq ... \leq W_\infty(\mu,\nu)$?
I stumbled upon this interesting statement in this paper:
"One interesting observation is that the Wasserstein ambiguity
set with the Wasserstein order p = 2 is less conservative, because the 2-...
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Understanding different norms in the p-Wasserstein distance
The generalized p-Wasserstein distance, for $p\geq 1$, is given by
$$d_W(Q_1,Q_2):=inf \left\{\int_{\Xi_2}||\xi_1-\xi_2||^p \Pi(d\xi_1,d\xi_2)\right\}$$
where $\Pi$ is the joint distribution of $\xi_1$...
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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 ...
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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. ...
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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,...
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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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Wasserstein distance inequality
Suppose $(\Omega, \mathcal F, \mathbb P)$ is a probability space. Suppose $X, X', Y, Y'$ are random variables.
Denote $W_1$ the Wasserstein-1 distance between $\mathbb P_X$ and $\mathbb P_{X'}$
and $...
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Wasserstein-1 distance, $W(A\cdot B \| C\cdot D)=W(A \| C) + W(B \| D)$
For 4 independent random variables $A, B, C, D$ and Wasserstein-1 distance $W^1$,
$W^1(P_{A,B} \parallel P_{C,D})=W^1(P_A \parallel P_C)+W^1(P_B \parallel P_D)$
Does the above equation generally ...
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Weighted median of distribution functions
I am working on the following barycenter problem: Suppose we are given $N>1$ probability measures on $\mathbb{R}$ with cumulative distribution functions $F_1,\dots,F_N$ and weights $a_1, \dots, a_N ...
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