<s>
The	O
Wasserstein	B-Algorithm
Generative	I-Algorithm
Adversarial	I-Algorithm
Network	I-Algorithm
(	O
WGAN	O
)	O
is	O
a	O
variant	O
of	O
generative	B-Algorithm
adversarial	I-Algorithm
network	I-Algorithm
(	O
GAN	O
)	O
proposed	O
in	O
2017	O
that	O
aims	O
to	O
"	O
improve	O
the	O
stability	O
of	O
learning	O
,	O
get	O
rid	O
of	O
problems	O
like	O
mode	O
collapse	O
,	O
and	O
provide	O
meaningful	O
learning	O
curves	O
useful	O
for	O
debugging	O
and	O
hyperparameter	O
searches	O
"	O
.	O
</s>
<s>
Compared	O
with	O
the	O
original	O
GAN	O
discriminator	O
,	O
the	O
Wasserstein	B-Algorithm
GAN	I-Algorithm
discriminator	O
provides	O
a	O
better	O
learning	O
signal	O
to	O
the	O
generator	O
.	O
</s>
<s>
The	O
Wasserstein	B-Algorithm
GAN	I-Algorithm
is	O
obtained	O
by	O
using	O
the	O
Wasserstein	O
metric	O
,	O
which	O
satisfies	O
a	O
"	O
dual	O
representation	O
theorem	O
"	O
that	O
renders	O
it	O
highly	O
efficient	O
to	O
compute	O
:	O
</s>
<s>
By	O
the	O
Kantorovich-Rubenstein	O
duality	O
,	O
the	O
definition	O
of	O
Wasserstein	B-Algorithm
GAN	I-Algorithm
is	O
clear	O
:	O
</s>
<s>
In	O
the	O
Wasserstein	B-Algorithm
GAN	I-Algorithm
game	O
,	O
the	O
discriminator	O
provides	O
a	O
better	O
gradient	O
than	O
in	O
the	O
GAN	O
game	O
.	O
</s>
<s>
For	O
Wasserstein	B-Algorithm
GAN	I-Algorithm
,	O
.	O
</s>
<s>
Let	O
be	O
parametrized	O
by	O
,	O
then	O
we	O
can	O
perform	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
by	O
using	O
two	O
unbiased	O
estimators	O
of	O
the	O
gradient:where	O
we	O
used	O
the	O
reparametrization	O
trick	O
.	O
</s>
<s>
Similarly	O
for	O
the	O
generator	O
in	O
Wasserstein	B-Algorithm
GAN	I-Algorithm
.	O
</s>
<s>
For	O
Wasserstein	B-Algorithm
GAN	I-Algorithm
,	O
has	O
gradient	O
1	O
almost	O
everywhere	O
,	O
while	O
for	O
GAN	O
,	O
has	O
flat	O
gradient	O
in	O
the	O
middle	O
,	O
and	O
steep	O
gradient	O
elsewhere	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
variance	O
for	O
the	O
estimator	O
in	O
GAN	O
is	O
usually	O
much	O
larger	O
than	O
that	O
in	O
Wasserstein	B-Algorithm
GAN	I-Algorithm
.	O
</s>
<s>
Consider	O
training	O
a	O
GAN	O
to	O
generate	O
ImageNet	B-General_Concept
,	O
a	O
collection	O
of	O
photos	O
of	O
size	O
256-by-256	O
.	O
</s>
<s>
The	O
space	O
of	O
all	O
such	O
photos	O
is	O
,	O
and	O
the	O
distribution	O
of	O
ImageNet	B-General_Concept
pictures	O
,	O
,	O
concentrates	O
on	O
a	O
manifold	O
of	O
much	O
lower	O
dimension	O
in	O
it	O
.	O
</s>
<s>
Training	O
the	O
generator	O
in	O
Wasserstein	B-Algorithm
GAN	I-Algorithm
is	O
just	O
gradient	B-Algorithm
descent	I-Algorithm
,	O
the	O
same	O
as	O
in	O
GAN	O
(	O
or	O
most	O
deep	O
learning	O
methods	O
)	O
,	O
but	O
training	O
the	O
discriminator	O
is	O
different	O
,	O
as	O
the	O
discriminator	O
is	O
now	O
restricted	O
to	O
have	O
bounded	O
Lipschitz	O
norm	O
.	O
</s>
<s>
Then	O
,	O
for	O
any	O
,	O
let	O
,	O
we	O
have	O
by	O
the	O
chain	O
rule:Thus	O
,	O
the	O
Lipschitz	O
norm	O
of	O
is	O
upper-bounded	O
bywhere	O
is	O
the	O
operator	O
norm	O
of	O
the	O
matrix	O
,	O
that	O
is	O
,	O
the	O
largest	O
singular	O
value	O
of	O
the	O
matrix	O
,	O
that	O
is	O
,	O
the	O
spectral	O
radius	O
of	O
the	O
matrix	O
(	O
these	O
concepts	O
are	O
the	O
same	O
for	O
matrices	O
,	O
but	O
different	O
for	O
general	O
linear	B-Architecture
operators	I-Architecture
)	O
.	O
</s>
