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Results tagged with optimal-transportation
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user 123897
2
votes
Accepted
Kantorovich duality with pseudometrics
Yes the Kantorovich Duality holds for continuous cost functions by following the proof in Villani's book without any change. The proof for general cost functions needs compactness of the set of coupli …
- 576
1
vote
Accepted
Is there a coupling that induces a given coupling via a transition kernel?
It's not true. If $\mu \mapsto \tilde\mu$ is injective and $X$ has at least two points then the transition kernels have to be deterministic, i.e. there is a measurable map $T:X\to Y$ such that $\kappa …
- 576
2
votes
Accepted
Monge-Kantorovich duality with a $\{0,1\}$ cost function
The strong Kantorovich duality, i.e. the existence of dual solutions, holds whenever $c$ is lower semicontinuous and real-valued, and the optimal coupling has finite cost, see Theorem 5.10(ii) in Vill …
- 576
3
votes
Effect of perturbing the atoms of a measure on the Wasserstein distance
Without further constraints this is not true and easy to see if $X$ is Euclidean: Let $\Gamma$ be the support of an optimal coupling between $G$ and $G'$. If for fixed $y\in p_2(\Gamma)$ the set $\{x …
- 576