Kantorovich problem in optimal transport
E1020368
The Kantorovich problem in optimal transport is a relaxed, linear-programming formulation of transporting mass between probability distributions that allows splitting mass and guarantees existence of optimal transport plans.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Kantorovich formulation of optimal transport | 1 |
| Monge–Kantorovich problem | 1 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
linear programming problem
ⓘ
mathematical optimization problem ⓘ relaxed optimal transport formulation ⓘ |
| allows | splitting of mass ⓘ |
| appliedIn |
economics
ⓘ
fluid mechanics ⓘ image processing ⓘ machine learning ⓘ statistics ⓘ |
| assumes | finite cost integral for admissible plans ⓘ |
| comparedTo | Monge problem without mass splitting ⓘ |
| constraint |
fixed marginals
ⓘ
nonnegative transport plan ⓘ |
| domain |
Polish spaces
NERFINISHED
ⓘ
metric measure spaces ⓘ probability measures ⓘ |
| dualObjective | maximize integral of potentials under marginal constraints ⓘ |
| dualVariables | Kantorovich potentials NERFINISHED ⓘ |
| ensures | existence of minimizers on compact spaces with lower semicontinuous cost ⓘ |
| equivalentTo | Wasserstein distance definition for suitable costs ⓘ |
| field |
mathematical analysis
ⓘ
operations research NERFINISHED ⓘ optimal transport theory ⓘ probability theory ⓘ |
| formulationType | primal linear program ⓘ |
| generalizes | Monge optimal transport problem NERFINISHED ⓘ |
| guarantees | existence of optimal transport plans under mild conditions ⓘ |
| hasDual | Kantorovich dual problem NERFINISHED ⓘ |
| introducedBy | Leonid Kantorovich NERFINISHED ⓘ |
| namedAfter | Leonid Kantorovich NERFINISHED ⓘ |
| objective | minimize expected transport cost ⓘ |
| property |
convex optimization problem
ⓘ
lower semicontinuity of cost functional under standard assumptions ⓘ |
| relatedConcept |
Earth mover's distance
NERFINISHED
ⓘ
Kantorovich–Rubinstein duality NERFINISHED ⓘ Wasserstein barycenter ⓘ entropic regularization of optimal transport ⓘ |
| relaxes | deterministic transport maps requirement ⓘ |
| solutionObject | optimal transport plan ⓘ |
| solutionSpace | space of probability measures on product space ⓘ |
| timePeriod | mid 20th century ⓘ |
| typicalCostFunction |
metric distance on underlying space
ⓘ
power of a metric distance ⓘ |
| usedToDefine |
Wasserstein-1 distance
NERFINISHED
ⓘ
Wasserstein-p distance NERFINISHED ⓘ |
| uses |
cost function
ⓘ
couplings of probability measures ⓘ transport plans ⓘ |
| yields | Wasserstein metric between probability measures ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Kantorovich formulation of optimal transport
this entity surface form:
Monge–Kantorovich problem