Questions tagged [optimal-transportation]
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260 questions
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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 ...
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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
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Modulus of of continuity of a convolution operator with respect to Wasserstein metric
For a (discrete) measure $G$ on some reasonable metric space $\Theta$, consider the map $G \mapsto f_G$ defined as
$$
f_G := f*G(dx) := \int f(dx|\theta) G(d\theta)
$$
for some nice kernel function $...
- 1,741
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Constructing a family of domains for application of method of continuity in optimal transportation
Anyway can help me about this paper?
http://arxiv.org/pdf/math/0601086v4.pdf
I want to ask page 20, The author want to construct a family of subdomain for using method of continuity. But I can't ...
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1-Wasserstein distance between two multivariate normal
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,...
- 1,256
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$X$ Polish geodesic implies $(P_2(X), W_2)$ geodesic
If $X,d$ is a complete and separable space then the space of Borel probability measures with finite second moment on $X$ endowed with the Wasserstein distance $W_2$ is geodesic.
I am looking for a ...
- 41
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a generalization of Monge-Kantorovich Problem
I am thinking about the martingale version of Monge-Kantorovich Problem.
Let $\mu(x)$ and $\nu(y)$ denote two density laws on $\mathbb{R}$, and define $M(\mu,\nu)$ the set of densities $f(x,y)$ on $\...
- 1,835
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Tangential boundary regularity for optimal transport maps
I'm interested in (and a bit confused by) the following theorem of Caffarelli, proven in section $4$ of his paper Boundary regularity of maps with convex potentials II:
Assume $u$ is a convex ...
- 4,899
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Where to use differential calculus on space of measures?
One great inside of Felix Otto is that the Wasserstein metric from optimal transportation gives the space of (finite second moment, probability) measures on $\mathbb{R}^n$ (or a manifold) a kind of ...
- 14.4k
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2-Wasserstein (optimal transport) and extension to the set of all signed measures
Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as
$$
d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}
$$
...
- 1,741