All Questions
Tagged with optimal-transportation pr.probability
114 questions
1
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176
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Rate of convergence of empirical distribution with respect to Wasserstein distance induced by binary cost function
Let $\mathcal X=(\mathcal X, d)$ be a Polish space (i.e complete metric space), and let $\Omega$ be a non-empty subset. Consider the binary cost function $c_\Omega$ on $\mathcal X^2$ defined by $c_\...
- 6,853
1
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81
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Empirical estimation of $\inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega)$, given i.i.d samples from $\mu$ and $\nu$
Let $\mathcal X$ be a Polish space and $\Omega \subseteq \mathcal X^2$ be open. Let $\mu$ and $\nu$ be probability measures, and consider the quantity $c_\Omega(\mu,\nu)$ defined by
$$
c_\Omega(\mu,\...
- 6,853
0
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75
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Optimizer of a semi-discrete optimal transport problem
Provided two probability distributions $\mu(dx)=\rho(x)dx$ and $\nu(dx)=\sum_{i=1}^n p_i\delta_{y_i}(dx)$ that are supported on some measurable set $\Omega\subset\mathbb R^d$, we consider the semi-...
user128095
2
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1
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309
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Density in Wasserstein space
I am wondering whether the following result is true:
Let $\mathcal W_p(\mathbb R^d)$ be the Wasserstein space of order $p$ and let $\eta$ and $\gamma$ be two probability measures in $\mathcal W_p(\...
- 325
5
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1
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2k
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Earth mover/Wasserstein distance between a pdf and an empirical distribution
This question is inspired by this much older question:
Convergence of an empirical distribution w.r.t. the Hellinger distance
Let $P$ be a continuous probability distribution on a compact subset of $...
- 51
2
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1
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210
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Controlling Mean Difference Between Product and Joint Distributions Using Optimal Transportation
Suppose we have nonindependent random variables $X \sim P$ and $Y \sim Q$, where $P$ and $Q$ denote their marginal distributions. We are interested in upper bounding
$$
|\mathbf{E}_{X, Y\sim P \...
- 1,127
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103
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Expectation of maximal Wasserstein distance between empirical distribution and a pdf
Let $P$ be a continuous probability distribution on $R^d$, $X$ the random variable $\sim P$, and $
\hat{X}$ be n i.i.d samples drawn according to $P$. We have another variable $\mu \in S^{d-1}$.
Do ...
- 109
3
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243
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Parametric distances on product spaces of measures
Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you.
Let $X$ be a topological ...
- 6,853
8
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1
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727
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continuity of the Boltzmann entropy in the Wasserstein metric
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
- 5,380
4
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589
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Optimal transport between two distributions in a Markov chain
In a previous question, given an ergodic Markov chain, I'm interesting in sampling as short a path as possible with prescribed distributions for its endpoints. In a comment, I propose that the ...
- 6,853
2
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0
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75
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existence of minimizer to the dual problem of a martingale optimal transport type problem
Let $\nu$ be a given probability measure on $\mathbb R^2$ and consider function of the following form:
$$L(f)(x_1,x_2)=\sup_{y\,=\,(y_1,y_2)\,\in\, \operatorname{Graph} (f)} \{ x_1 y_2 + x_2 y_1 - ...
- 325
4
votes
1
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444
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PDE-Based Triangle Inequality for Optimal Transportation
Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and $\...
- 705
1
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121
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$\mathbb{P}(d(X,Y)>\alpha)<\beta$ if $\mathbb{P}(X\in E)\leq \mathbb{P}(Y\in E^{\alpha})+\beta$ for all measurable E
Given two random variables X,Y with measures P,Q. Show that if $P(E) \le Q(E^\alpha) + \beta$ for all measurable $E\subset\mathbb{R}$ then $\mathbb{P}(d(X,Y)>\alpha)<\beta$.
Only hints please.
...
- 395
0
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0
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184
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Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380