Brenier map
E1017920
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.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
optimal transport map ⓘ |
| appearsIn | polar factorization of vector-valued maps ⓘ |
| hasCodomain | Euclidean space NERFINISHED ⓘ |
| hasDomain | Euclidean space NERFINISHED ⓘ |
| hasRegularityDependingOn | regularity of source and target densities GENERATED ⓘ |
| is |
almost everywhere uniquely defined
ⓘ
gradient of a convex function ⓘ optimal transport map for quadratic cost ⓘ |
| isAssociatedWith |
L2 optimal transport
ⓘ
Wasserstein-2 distance ⓘ |
| isCharacterizedAs |
Monge solution of the quadratic optimal transport problem
ⓘ
gradient of a convex potential ⓘ |
| isDefinedBetween |
absolutely continuous probability measures
ⓘ
probability measures ⓘ |
| isDefinedUnder | quadratic cost ⓘ |
| isGeneralizedBy | optimal maps for other cost functions ⓘ |
| isGradientOf | convex potential function ⓘ |
| isGuaranteedBy | Brenier’s polar factorization theorem NERFINISHED ⓘ |
| isPushforwardMapOf | convex potential gradient ⓘ |
| isRelatedTo |
Kantorovich formulation of optimal transport
NERFINISHED
ⓘ
Monge formulation of optimal transport NERFINISHED ⓘ cyclical monotonicity ⓘ |
| isSolutionOf | Monge problem with quadratic cost ⓘ |
| isSpecialCaseOf | c-convex potential gradients for c(x,y)=|x−y|^2/2 ⓘ |
| isStableUnder | weak convergence of measures (under suitable conditions) ⓘ |
| isSubjectOf | regularity theory in optimal transport ⓘ |
| isToolIn |
fluid mechanics
ⓘ
geometric analysis ⓘ machine learning ⓘ modern optimal transport theory ⓘ partial differential equations ⓘ probability theory ⓘ |
| isUnique | up to source-measure null sets ⓘ |
| isUsedFor |
density equalization mappings
ⓘ
domain morphing and mesh transport ⓘ image registration in imaging sciences ⓘ measure-preserving rearrangements ⓘ |
| isUsedToDefine |
displacement interpolation
ⓘ
geodesics in Wasserstein-2 space ⓘ |
| minimizes | quadratic transport cost ⓘ |
| namedAfter | Yann Brenier NERFINISHED ⓘ |
| pushesForward | source measure to target measure ⓘ |
| requiresAssumption | source measure absolutely continuous with respect to Lebesgue measure ⓘ |
| satisfies |
Monge–Ampère equation in suitable settings
ⓘ
cyclical monotonicity of its graph ⓘ |
| underlies | Riemannian structure of Wasserstein space ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.