Kantorovich duality
E1017918
Kantorovich duality is a fundamental result in optimal transport theory that characterizes the optimal transport cost as the supremum of a dual variational problem over suitable test functions.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
duality theorem
ⓘ
result in optimal transport theory ⓘ |
| appliesTo |
Borel probability measures
ⓘ
Monge–Kantorovich optimal transport problem NERFINISHED ⓘ probability measures on Polish spaces ⓘ |
| assumes |
Polish or compact metric spaces in standard theorems
ⓘ
tightness of probability measures in many formulations ⓘ |
| characterizes | optimal transport cost ⓘ |
| expresses | optimal transport cost as supremum over dual potentials ⓘ |
| field |
convex analysis
ⓘ
linear programming ⓘ mathematical analysis ⓘ optimal transport ⓘ probability theory ⓘ |
| foundationFor |
Kantorovich–Rubinstein theorem
NERFINISHED
ⓘ
modern optimal transport theory ⓘ |
| generalizes | linear programming duality for transport problems ⓘ |
| hasDualFormulation | maximization over pairs of functions bounded by cost ⓘ |
| hasPrimalFormulation | minimization of transport cost over couplings GENERATED ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies |
equality of primal and dual optimal values
ⓘ
existence of optimal transport plans under mild conditions ⓘ |
| involves |
1-Lipschitz functions in Wasserstein-1 case
ⓘ
c-concave functions ⓘ dual potentials ⓘ |
| isRelatedTo |
Monge formulation of optimal transport
NERFINISHED
ⓘ
Wasserstein distances NERFINISHED ⓘ Wasserstein-1 distance ⓘ Wasserstein-p distances NERFINISHED ⓘ |
| isSpecialCaseOf | Fenchel–Rockafellar duality NERFINISHED ⓘ |
| isUsedIn |
Wasserstein GANs
NERFINISHED
ⓘ
economics ⓘ generative adversarial networks ⓘ gradient flows in Wasserstein space ⓘ image processing ⓘ machine learning ⓘ metric geometry of probability measures ⓘ partial differential equations ⓘ shape analysis ⓘ statistics ⓘ |
| namedAfter | Leonid Kantorovich NERFINISHED ⓘ |
| relates |
dual variational problem
ⓘ
primal optimal transport problem ⓘ |
| requires |
integrable cost function
ⓘ
lower semicontinuous cost function ⓘ |
| uses | test functions ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.