Monge problem in optimal transport
E326555
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Monge problem in optimal transport canonical | 1 |
| Monge–Kantorovich optimal transport problem | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical optimization problem
ⓘ
problem in optimal transport theory ⓘ |
| assumes | conservation of mass ⓘ |
| constraint | pushforward of source measure by T equals target measure ⓘ |
| contrastedWith | Kantorovich problem in optimal transport ⓘ |
| coreQuestion | find a transport map pushing one distribution to another with minimal cost ⓘ |
| difficulty |
may lack solutions for given marginals and cost
ⓘ
optimization problem is highly non-linear ⓘ |
| domain |
Euclidean space
ⓘ
surface form:
Euclidean spaces
Polish spaces ⓘ |
| field |
applied mathematics
ⓘ
calculus of variations ⓘ mathematical analysis ⓘ measure theory ⓘ optimal transport ⓘ probability theory ⓘ |
| formulatedBy | Gaspard Monge ⓘ |
| generalizationOf | classical earth mover problem ⓘ |
| hasConditionForExistence | absolute continuity of source measure for quadratic cost ⓘ |
| hasConditionForUniqueness | strict convexity of cost function ⓘ |
| hasContinuousVersion | Monge problem on continuous probability measures ⓘ |
| hasDiscreteVersion | Monge problem on finite point sets ⓘ |
| hasRelaxation |
Kantorovich problem in optimal transport
ⓘ
surface form:
Kantorovich formulation of optimal transport
|
| historicalSignificance | earliest formal statement of mass transportation problem ⓘ |
| inspired | development of Kantorovich duality ⓘ |
| involves |
cost function
ⓘ
source measure ⓘ target measure ⓘ transport map ⓘ |
| mathematicalFormulation | minimize integral of cost of x to T(x) over source measure ⓘ |
| namedAfter | Gaspard Monge ⓘ |
| relatedConcept |
Brenier map
ⓘ
Monge–Ampère equation ⓘ Wasserstein distance ⓘ mass transportation theory ⓘ |
| requires | deterministic transport map ⓘ |
| solutionRegularity | linked to regularity of Monge–Ampère type equations ⓘ |
| solutionType | optimal transport map ⓘ |
| typicalCostFunction |
Euclidean distance
ⓘ
general metric cost ⓘ squared Euclidean distance ⓘ |
| usedIn |
economics
ⓘ
fluid mechanics ⓘ geometry ⓘ image processing ⓘ machine learning ⓘ partial differential equations ⓘ |
| yearProposed | 1781 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Monge–Kantorovich optimal transport problem