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pr.chance – How to construe couplings in optimum exaltation? Answer

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pr.chance – How to construe couplings in optimum exaltation?

Let $mu$ and $nu$ breathe two measures on some (not less than measurable) house $X$. In optimum exaltation concept, Monge’s downside to
$$ textual content{decrease} quad int c(x,T(x))mu(dx) quad textual content{over measurable mappings }T: X rightarrow Y textual content{ and } T_#mu = nu$$
has a comparatively easy interpretation: We attempt to discover a measurable map $T$ that minimizes the expense to strike mass from $x$ to $T(x)$. Now, the Kantorovich downside to
$$ textual content{decrease} quad int c(x,y)pi(dx,dy) quad textual content{over couplings } pi textual content{ with first and second marginals } mu textual content{ and } nu textual content{, respectively,}$$
I discover to breathe mighty more durable to construe as a actual `mass switch’ downside; If $pi^star$ is an optimum coupling to the Kantorovich downside, what does $pi^star$ inform me the place how mighty mass actually goes? How do I construe the Kantorovich downside?

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