Mathematics Stack Exchange
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Questions tagged [wasserstein]

For questions related to the Wasserstein metric and the corresponding metric space. Also consider optimal-transport.

The Wasserstein metric on the space of probability measures on a metric space with certain finite moments is fundamental to optimal transport. Imaging the probability measures as piles of dirt, the Wasserstein metric between them is the amount of work (mass x distance) needed to transport one onto the other.

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weak convergence of JKO scheme

I was looking at the Theorem of the seminal paper by Jordan "The variational formulation of the Fokker-Planck equation" and in their "Theorem" in section 5 I was a bit confused by their setup. As I understood it, they are looking to prove weak…
Math_Day
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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 densities by the Wasserstein distance of the probability…
Fabian P
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Are U-statistics dense?

Consider the set $E$ of compactly supported Borel probability measures on the real line equipped with the Wasserstein infinity metric. A continuous function $f:E\to\mathbb R$ is called a U-statistic of degree $n$ if there's a measurable function…
Derivative
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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 of $\|f-g\|_{L^p}$ (especially for $p=1$ and…
Fei Cao
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Advection reaction equation is solved by projection of solution of continuity equation

Suppose an absolutely continuous curve $\mu \colon (0, \infty) \to P_2(\Omega)$, where $P_2$ is the Wasserstein-2-space, fulfils the continuity equation $$ \label{eq:CE} \tag{CE} \partial_t \mu_t = \text{div}(\mu_t g_{\mathfrak h \mu_t}) $$ for…
ViktorStein
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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\sum_iX_i \right|\geq t\right)\leq…
Tyler6
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Maximiser of $W_1(\mu, \nu)$ can be changed outside of $\text{conv}(\text{supp}(\mu) \cap \text{supp}(\nu))$ (under additional assumptions)

Let $(X, \| \cdot \|)$ be a reflexive Banach space and $\mathbb{P}_n$, $\mathbb{P}_r$ be measures on $X$. Let the support of $\mathbb{P}_r$, $M := \text{supp}(\mathbb{P}_r)$ be a weakly compact set and $$P_M \colon D \to M, \qquad x \mapsto…
ViktorStein
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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 \Omega $ be continuous. Let $\nu\in P(\Omega)$ be a…
John
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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 $\mu_1$, $\mu_2 \in \mathcal{P}_2(\mathbb{R}^{n})$ and $\varepsilon >0$, can…
mnmn1993
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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 $$ d_K(X,Y)=\underset{x \in \mathbb{R}}{sup}\{|F_X(x) - F_Y(x)|\} $$ where $F_X(x)$ is the comulative distribution…
fabianod
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Relation between Wasserstein distance and distribution convergence

Let's have a succession $X_n$ of real value random variables and another real value random variable X, then $$ X_n \xrightarrow{d} X \iff \lim_{n \to \infty}{}d_K(X_n,X) = 0, $$ where $d_K(X,Y)$ is the Kolmogorov-Smirnov distance between X and…
fabianod
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Scaling property of the Wasserstein metric

I would need help with this example. Let $(S, ||\cdot||)$ denote a normed vector space over $K =\mathbb R$ or $K =\mathbb C$. Let $X$ and $Y$ be $S$-valued random vectors with $E~[~||X||~] < \infty$ and $E~[~||Y||~] < \infty$. Prove that, for every…
Spira
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Optimal Transport between two Gaussians

Consider the optimal transport map $T$ between $N(\mu_0,\Sigma_0)$ and $N(\mu_1,\Sigma_1)$. I believed that the optimal transport was given by: $$ T(x) = \mu_1 + \Sigma_1^{1/2} \Sigma_0^{-1/2}(x-\mu_0) $$ However in Peyre's book "Computational…
Kevin Ro
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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(x, x_0) \ d \nu < \infty \end{equation} for some…
Kakuro
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Wasserstein metric vs Holder continuity

It is well known that if $f$ is a Lipschitz continuous function, i.e. $$\forall x,y\in \Omega\qquad |f(x)-f(y)|\le L\|x-y\|$$ then, for any two probability distributions $\mu, \nu$ $$\int_\Omega f(x)(d\mu - d\nu) \le LW(\mu,\nu)$$ where $W(\mu,\nu)$…
Davide Maran
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