Affiliation:
1. Dipartimento di Matematica , Università di Pisa , Largo B. Pontecorvo 5, 56127 Pisa , Italy
2. CEREMADE UMR CNRS 7534 , Université de Paris Dauphine , Place de Lattre de Tassigny, 75775 Paris Cedex 16 , France
3. Department of Mathematics and Statistics , McGill University , 805 Rue Sherbrooke Ouest , Montréal , Canada
Abstract
Abstract
We study the Wasserstein distance between two measures
μ
,
ν
{\mu,\nu}
which are mutually singular. In particular, we are interested in minimization problems of the form
W
(
μ
,
𝒜
)
=
inf
{
W
(
μ
,
ν
)
:
ν
∈
𝒜
}
,
W(\mu,\mathcal{A})=\inf\{W(\mu,\nu):\nu\in\mathcal{A}\},
where μ is a given probability and
𝒜
{\mathcal{A}}
is contained in the class
μ
⊥
{\mu^{\perp}}
of probabilities that are singular with respect to μ. Several cases for
𝒜
{\mathcal{A}}
are considered; in particular, when
𝒜
{\mathcal{A}}
consists of
L
1
{L^{1}}
densities bounded by a constant, the optimal solution is given by the characteristic function of a domain. Some regularity properties of these optimal domains are also studied. Some numerical simulations are included, as well as the double minimization problem
min
{
P
(
B
)
+
k
W
(
A
,
B
)
:
|
A
∩
B
|
=
0
,
|
A
|
=
|
B
|
=
1
}
,
\min\{P(B)+kW(A,B):|A\cap B|=0,\,|A|=|B|=1\},
where
k
>
0
{k>0}
is a fixed constant,
P
(
A
)
{P(A)}
is the perimeter of A, and both sets
A
,
B
{A,B}
may vary.
Funder
Agence Nationale de la Recherche
Subject
Applied Mathematics,Analysis
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