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Bounty Ended with Nawaf Bou-Rabee's answer chosen by ViktorStein
the question is about the Wasserstein distance induced by the underlying distance $c$
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Nawaf Bou-Rabee
Nawaf Bou-Rabee
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Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{x_i}$ with $\nu_1,\ldots,\nu_n \ge 0$ and $\sum_i \nu_i = 1$.

Let $(x_i,x_j) \mapsto c(x_i,x_j)$ be a distance on the $n$ points $x_1,\ldots,x_n$. For example, $c(x_i,x_j) := \|x_i-x_j\|_p$ for some $p \in [1,\infty)$.

Question

What's a closed form formula for the corresponding Wasserstein distance between $\mu$ and $\nu$ ?

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{x_i}$ with $\nu_1,\ldots,\nu_n \ge 0$ and $\sum_i \nu_i = 1$.

Let $(x_i,x_j) \mapsto c(x_i,x_j)$ be a distance on the $n$ points $x_1,\ldots,x_n$. For example, $c(x_i,x_j) := \|x_i-x_j\|_p$ for some $p \in [1,\infty)$.

Question

What's a closed form formula for the Wasserstein distance between $\mu$ and $\nu$ ?

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{x_i}$ with $\nu_1,\ldots,\nu_n \ge 0$ and $\sum_i \nu_i = 1$.

Let $(x_i,x_j) \mapsto c(x_i,x_j)$ be a distance on the $n$ points $x_1,\ldots,x_n$. For example, $c(x_i,x_j) := \|x_i-x_j\|_p$ for some $p \in [1,\infty)$.

Question

What's a closed form formula for the corresponding Wasserstein distance between $\mu$ and $\nu$ ?

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edit approved May 23, 2022 at 13:26
ViktorStein
edit approved May 23, 2022 at 13:26
ViktorStein
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minor typo
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JHM
JHM
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Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{x_i}$ with $\nu_1,\ldots,\nu_n \ge 0$ and $\sum_i \nu_i = 1$.

Let $(x_i,x_j) \mapsto c(x_i,y_j)$$(x_i,x_j) \mapsto c(x_i,x_j)$ be a distance on the $n$ points $x_1,\ldots,x_n$. For example, $c(x_i,x_j) := \|x_i-x_j\|_p$ for some $p \in [1,\infty)$.

Question

What's a closed form formula for the Wasserstein distance between $\mu$ and $\nu$ ?

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{x_i}$ with $\nu_1,\ldots,\nu_n \ge 0$ and $\sum_i \nu_i = 1$.

Let $(x_i,x_j) \mapsto c(x_i,y_j)$ be a distance on the $n$ points $x_1,\ldots,x_n$. For example, $c(x_i,x_j) := \|x_i-x_j\|_p$ for some $p \in [1,\infty)$.

Question

What's a closed form formula for the Wasserstein distance between $\mu$ and $\nu$ ?

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{x_i}$ with $\nu_1,\ldots,\nu_n \ge 0$ and $\sum_i \nu_i = 1$.

Let $(x_i,x_j) \mapsto c(x_i,x_j)$ be a distance on the $n$ points $x_1,\ldots,x_n$. For example, $c(x_i,x_j) := \|x_i-x_j\|_p$ for some $p \in [1,\infty)$.

Question

What's a closed form formula for the Wasserstein distance between $\mu$ and $\nu$ ?

Notice added Draw attention by ViktorStein
Bounty Started worth 50 reputation by ViktorStein
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dohmatob
dohmatob
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