<s>
In	O
machine	O
learning	O
,	O
a	O
variational	B-Algorithm
autoencoder	I-Algorithm
(	O
VAE	O
)	O
,	O
is	O
an	O
artificial	B-Architecture
neural	I-Architecture
network	I-Architecture
architecture	O
introduced	O
by	O
Diederik	O
P	O
.	O
Kingma	O
and	O
Max	O
Welling	O
,	O
belonging	O
to	O
the	O
families	O
of	O
probabilistic	O
graphical	O
models	O
and	O
variational	O
Bayesian	O
methods	O
.	O
</s>
<s>
Variational	B-Algorithm
autoencoders	I-Algorithm
are	O
often	O
associated	O
with	O
the	O
autoencoder	B-Algorithm
model	O
because	O
of	O
its	O
architectural	O
affinity	O
,	O
but	O
with	O
significant	O
differences	O
in	O
the	O
goal	O
and	O
mathematical	O
formulation	O
.	O
</s>
<s>
Variational	B-Algorithm
autoencoders	I-Algorithm
are	O
probabilistic	O
generative	O
models	O
that	O
require	O
neural	B-Architecture
networks	I-Architecture
as	O
only	O
a	O
part	O
of	O
their	O
overall	O
structure	O
.	O
</s>
<s>
The	O
neural	B-Architecture
network	I-Architecture
components	O
are	O
typically	O
referred	O
to	O
as	O
the	O
encoder	O
and	O
decoder	O
for	O
the	O
first	O
and	O
second	O
component	O
respectively	O
.	O
</s>
<s>
The	O
first	O
neural	B-Architecture
network	I-Architecture
maps	O
the	O
input	O
variable	O
to	O
a	O
latent	O
space	O
that	O
corresponds	O
to	O
the	O
parameters	O
of	O
a	O
variational	O
distribution	O
.	O
</s>
<s>
Although	O
this	O
type	O
of	O
model	O
was	O
initially	O
designed	O
for	O
unsupervised	B-General_Concept
learning	I-General_Concept
,	O
its	O
effectiveness	O
has	O
been	O
proven	O
for	O
semi-supervised	B-General_Concept
learning	I-General_Concept
and	O
supervised	B-General_Concept
learning	I-General_Concept
.	O
</s>
<s>
A	O
variational	B-Algorithm
autoencoder	I-Algorithm
is	O
a	O
generative	O
model	O
with	O
a	O
prior	O
and	O
noise	O
distribution	O
respectively	O
.	O
</s>
<s>
However	O
,	O
variational	B-Algorithm
autoencoders	I-Algorithm
use	O
a	O
neural	B-Architecture
network	I-Architecture
as	O
an	O
amortized	O
approach	O
to	O
jointly	O
optimize	O
across	O
data	O
points	O
.	O
</s>
<s>
This	O
neural	B-Architecture
network	I-Architecture
takes	O
as	O
input	O
the	O
data	O
points	O
themselves	O
,	O
and	O
outputs	O
parameters	O
for	O
the	O
variational	O
distribution	O
.	O
</s>
<s>
The	O
decoder	O
is	O
the	O
second	O
neural	B-Architecture
network	I-Architecture
of	O
this	O
model	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
use	O
another	O
neural	B-Architecture
network	I-Architecture
that	O
maps	O
to	O
the	O
variance	O
,	O
however	O
this	O
can	O
be	O
omitted	O
for	O
simplicity	O
.	O
</s>
<s>
In	O
such	O
a	O
case	O
,	O
the	O
variance	O
can	O
be	O
optimized	O
with	O
gradient	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
In	O
the	O
vanilla	O
variational	B-Algorithm
autoencoder	I-Algorithm
,	O
is	O
usually	O
taken	O
to	O
be	O
a	O
finite-dimensional	O
vector	O
of	O
real	O
numbers	O
,	O
and	O
to	O
be	O
a	O
Gaussian	O
distribution	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
the	O
problem	O
is	O
to	O
find	O
a	O
good	O
probabilistic	O
autoencoder	B-Algorithm
,	O
in	O
which	O
the	O
conditional	O
likelihood	O
distribution	O
is	O
computed	O
by	O
the	O
probabilistic	O
decoder	O
,	O
and	O
the	O
approximated	O
posterior	O
distribution	O
is	O
computed	O
by	O
the	O
probabilistic	O
encoder	O
.	O
</s>
<s>
As	O
in	O
every	O
deep	B-Algorithm
learning	I-Algorithm
problem	O
,	O
it	O
is	O
necessary	O
to	O
define	O
a	O
differentiable	O
loss	O
function	O
in	O
order	O
to	O
update	O
the	O
network	O
weights	O
through	O
backpropagation	B-Algorithm
.	O
</s>
<s>
For	O
variational	B-Algorithm
autoencoders	I-Algorithm
,	O
the	O
idea	O
is	O
to	O
jointly	O
optimize	O
the	O
generative	O
model	O
parameters	O
to	O
reduce	O
the	O
reconstruction	O
error	O
between	O
the	O
input	O
and	O
the	O
output	O
,	O
and	O
to	O
make	O
as	O
close	O
as	O
possible	O
to	O
.	O
</s>
<s>
As	O
reconstruction	O
loss	O
,	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
and	O
cross	O
entropy	O
are	O
often	O
used	O
.	O
</s>
<s>
To	O
efficiently	O
search	O
for	O
the	O
typical	O
method	O
is	O
gradient	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
The	O
reparameterization	O
trick	O
(	O
also	O
known	O
as	O
stochastic	O
backpropagation	B-Algorithm
)	O
bypasses	O
this	O
difficulty	O
.	O
</s>
<s>
Here	O
,	O
is	O
obtained	O
by	O
the	O
Cholesky	O
decomposition:Then	O
we	O
haveand	O
so	O
we	O
obtained	O
an	O
unbiased	O
estimator	O
of	O
the	O
gradient	O
,	O
allowing	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
Many	O
variational	B-Algorithm
autoencoders	I-Algorithm
applications	O
and	O
extensions	O
have	O
been	O
used	O
to	O
adapt	O
the	O
architecture	O
to	O
other	O
domains	O
and	O
improve	O
its	O
performance	O
.	O
</s>
<s>
Some	O
structures	O
directly	O
deal	O
with	O
the	O
quality	O
of	O
the	O
generated	O
samples	O
or	O
implement	O
more	O
than	O
one	O
latent	O
space	O
to	O
further	O
improve	O
the	O
representation	B-General_Concept
learning	I-General_Concept
.	O
</s>
<s>
Some	O
architectures	O
mix	O
VAE	O
and	O
generative	B-Algorithm
adversarial	I-Algorithm
networks	I-Algorithm
to	O
obtain	O
hybrid	O
models	O
.	O
</s>
