Answer by Michael Greinecker for How to interpret couplings in optimal...
The interpretation of couplings as randomized transport maps is not without problems. In general, extreme points of the space of couplings need not be supported on the graph of a function; see, for...
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Answer by Gabe K for How to interpret couplings in optimal transport?
Given any coupling $\pi$ between two measures $\mu$ and $\nu$, you can construct aassociated transport plan. To do so, for each point $x \in X$, you distribute the mass according the "marginal...
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Answer by Gilles Mordant for How to interpret couplings in optimal transport?
The answers above already say it all.Still, if you are a visual person, have a look at the following optimal coupling/transport plan (or to be precise, a sample thereof. Note that the colour's strength...
View ArticleAnswer by Yuval Peres for How to interpret couplings in optimal transport?
Of the mass $\mu(A)$ in $A$ a fraction $\pi(A \times B)$ is transported to $B$, so you can think of this as a randomized transport map. A basic example to think of is $\mu=\delta_0$ and...
View ArticleHow to interpret couplings in optimal transport?
Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to$$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable...
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