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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kantorovich duality canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T13035697 — 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 duality Context triple: [Monge–Ampère equation, relatedTo, Kantorovich duality]
-
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.
Karush–Kuhn–Tucker conditions
The Karush–Kuhn–Tucker conditions are fundamental optimality criteria in nonlinear programming that generalize Lagrange multipliers to handle inequality constraints.
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D.
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.
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E.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
- 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 duality Target entity description: 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.
-
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.
Karush–Kuhn–Tucker conditions
The Karush–Kuhn–Tucker conditions are fundamental optimality criteria in nonlinear programming that generalize Lagrange multipliers to handle inequality constraints.
-
D.
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.
-
E.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Kantorovich duality Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.