In optimal transport, Brenier's theorem is a theorem about the optimal solution to a transportation problem on Euclidean space. It states that the optimal transportation plan of an absolutely continuous probability measure is the gradient of a convex function.
More precisely, if
μ
{\displaystyle \mu }
and
ν
{\displaystyle \nu }
are probability measures on
R
n
{\displaystyle \mathbb {R} ^{n}}
with finite second moments and
μ
{\displaystyle \mu }
is absolutely continuous with respect to Lebesgue measure, then there is a unique optimal transport map
T
{\displaystyle T}
pushing
μ
{\displaystyle \mu }
forward to
ν
{\displaystyle \nu }
for the cost
|
x
−
y
|
2
{\displaystyle |x-y|^{2}}
. This map has the form
T
(
x
)
=
∇
φ
(
x
)
{\displaystyle T(x)=\nabla \varphi (x)}
for a convex function
φ
:
R
n
→
(
−
∞
,
+
∞
]
{\displaystyle \varphi :\mathbb {R} ^{n}\to (-\infty ,+\infty ]}
, uniquely determined up to changes that do not affect its gradient on the support of
μ
{\displaystyle \mu }
.
The theorem identifies convex gradients as the higher-dimensional analogue of increasing rearrangements on the real line. In one dimension, the optimal way to transport one probability distribution to another for quadratic cost is the monotone rearrangement. In higher dimensions there is no natural total ordering of points, and Brenier's theorem replaces monotonicity by cyclic monotonicity, which is characterized by gradients of convex functions.
Brenier's theorem is closely related to the polar factorization theorem, also due to Yann Brenier, which decomposes a suitable vector field as the composition of a measure-preserving map and the gradient of a convex function.
Contents
Statement
Let
P
2
(
R
n
)
{\displaystyle {\mathcal {P}}_{2}(\mathbb {R} ^{n})}
denote the set of Borel probability measures on
R
n
{\displaystyle \mathbb {R} ^{n}}
with finite second moment. If
μ
,
ν
∈
P
2
(
R
n
)
{\displaystyle \mu ,\nu \in {\mathcal {P}}_{2}(\mathbb {R} ^{n})}
, a measurable map
T
:
R
n
→
R
n
{\displaystyle T:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}
Inverse map
If both
μ
{\displaystyle \mu }
and
ν
{\displaystyle \nu }
are absolutely continuous, then the Brenier map from
μ
{\displaystyle \mu }
to
ν
{\displaystyle \nu }
has an inverse in the almost-everywhere sense. If
T
=
∇
φ
{\displaystyle T=\nabla \varphi }
is the Brenier map from
μ
{\displaystyle \mu }
to
ν
{\displaystyle \nu }
, then the Brenier map from
ν
{\displaystyle \nu }
back to
μ
{\displaystyle \mu }