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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Kantorovich formulation of optimal transport | 2 |
| Kantorovich problem in optimal transport canonical | 1 |
| Monge–Kantorovich problem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T13035724 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Kantorovich problem in optimal transport Context triple: [Monge problem in optimal transport, contrastedWith, Kantorovich problem in optimal transport]
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A.
Monge problem in optimal transport
The Monge problem in optimal transport is a foundational mathematical formulation that seeks the most efficient way to move mass from one distribution to another, minimizing a given transportation cost.
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B.
Optimal Transport: Old and New
"Optimal Transport: Old and New" is a comprehensive monograph by Cédric Villani that develops the theory of optimal transport and its applications across analysis, geometry, and probability.
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C.
Kantorovich duality
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.
-
D.
Brenier map
The Brenier map is the unique gradient of a convex function that provides the optimal transport between probability measures under a quadratic cost, playing a central role in modern optimal transport theory.
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E.
Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Kantorovich problem in optimal transport Target entity description: 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.
-
A.
Monge problem in optimal transport
The Monge problem in optimal transport is a foundational mathematical formulation that seeks the most efficient way to move mass from one distribution to another, minimizing a given transportation cost.
-
B.
Optimal Transport: Old and New
"Optimal Transport: Old and New" is a comprehensive monograph by Cédric Villani that develops the theory of optimal transport and its applications across analysis, geometry, and probability.
-
C.
Kantorovich duality
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.
-
D.
Brenier map
The Brenier map is the unique gradient of a convex function that provides the optimal transport between probability measures under a quadratic cost, playing a central role in modern optimal transport theory.
-
E.
Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
- F. None of above. chosen
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
ⓘ
metric measure spaces ⓘ probability measures ⓘ |
| dualObjective | maximize integral of potentials under marginal constraints ⓘ |
| dualVariables |
Kantorovich potentials
ⓘ
linked to:
Kantorovich duality
|
| ensures | existence of minimizers on compact spaces with lower semicontinuous cost ⓘ |
| equivalentTo | Wasserstein distance definition for suitable costs ⓘ |
| field |
mathematical analysis
ⓘ
operations research ⓘ optimal transport theory ⓘ probability theory ⓘ |
| formulationType | primal linear program ⓘ |
| generalizes |
Monge optimal transport problem
ⓘ
linked to:
Monge problem in optimal transport
|
| guarantees | existence of optimal transport plans under mild conditions ⓘ |
| hasDual |
Kantorovich dual problem
ⓘ
linked to:
Kantorovich duality
|
| introducedBy |
Leonid Kantorovich
ⓘ
linked to:
Tjalling’s co-laureate Leonid Kantorovich
|
| namedAfter |
Leonid Kantorovich
ⓘ
linked to:
Tjalling’s co-laureate Leonid Kantorovich
|
| objective | minimize expected transport cost ⓘ |
| property |
convex optimization problem
ⓘ
lower semicontinuity of cost functional under standard assumptions ⓘ |
| relatedConcept |
Earth mover's distance
ⓘ
linked to:
Earth Mover's distance
Kantorovich–Rubinstein duality ⓘ
linked to:
Kantorovich duality
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
ⓘ
linked to:
Wasserstein distance
Wasserstein-p distance ⓘ
linked to:
Wasserstein distance
|
| uses |
cost function
ⓘ
couplings of probability measures ⓘ transport plans ⓘ |
| yields | Wasserstein metric between probability measures ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Kantorovich problem in optimal transport Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Kantorovich problem in optimal transport
linked to: Kantorovich problem in optimal transport
linked to: Kantorovich problem in optimal transport