Wasserstein
Measure
If
P
and
Q
are
two
probability
distributions,
then
Wasserstein
distance
is
defined
as
W(PllQ)
=
__inf
'Eq
.y
llz—yll]
J_GH(P,@
P('j):
r('x,
)J)\.
«
II(P,
Q)
denotes
the
set
of
all
joint
distributions
y
(x,
y)
whose
marginals
are
P
and
Q
respectively
*
y(x,y)
represents
the
"mass"
transported
from
x
(in
distribution
P)
to
y (in
distribution
Q)
*
The
Wasserstein
distance
(Earth
mover
distance)
is
minimum
cost
of
undertaking
such
a
transport
The
Wasserstein
distance
has
a
dual
form
(Kantorovich-Rubinstein):
W
(Pr,
Pg)
=
SUP<
]EmNPr@
—
Eqnpy
[
()]
Supremum
over
W
P/
8
)
HlfllL-i
o
K-Lipschitz
=—
sup
E,op
[f(2)]
—Egnp,[f(z)]
([
Tunctions
fllL<K
M
M
7Y
€
R
,f
IR
=
where,
f(.)
is
K-Lipschitz
if"f(x)
—fI
<K|x—y]|
N
N=4
Arjovsky
et
al.,
Wasserstein
GAN,
2017