<s>
A	O
restricted	B-Algorithm
Boltzmann	I-Algorithm
machine	I-Algorithm
(	O
RBM	O
)	O
is	O
a	O
generative	O
stochastic	O
artificial	B-Architecture
neural	I-Architecture
network	I-Architecture
that	O
can	O
learn	O
a	O
probability	O
distribution	O
over	O
its	O
set	O
of	O
inputs	O
.	O
</s>
<s>
RBMs	O
have	O
found	O
applications	O
in	O
dimensionality	B-Algorithm
reduction	I-Algorithm
,	O
</s>
<s>
classification	B-General_Concept
,	O
</s>
<s>
collaborative	B-Algorithm
filtering	I-Algorithm
,	O
feature	B-General_Concept
learning	I-General_Concept
,	O
</s>
<s>
and	O
even	O
many	B-Algorithm
body	I-Algorithm
quantum	I-Algorithm
mechanics	I-Algorithm
.	O
</s>
<s>
They	O
can	O
be	O
trained	O
in	O
either	O
supervised	B-General_Concept
or	O
unsupervised	B-General_Concept
ways	O
,	O
depending	O
on	O
the	O
task	O
.	O
</s>
<s>
As	O
their	O
name	O
implies	O
,	O
RBMs	O
are	O
a	O
variant	O
of	O
Boltzmann	B-Algorithm
machines	I-Algorithm
,	O
with	O
the	O
restriction	O
that	O
their	O
neurons	O
must	O
form	O
a	O
bipartite	O
graph	O
:	O
</s>
<s>
By	O
contrast	O
,	O
"	O
unrestricted	O
"	O
Boltzmann	B-Algorithm
machines	I-Algorithm
may	O
have	O
connections	O
between	O
hidden	O
units	O
.	O
</s>
<s>
This	O
restriction	O
allows	O
for	O
more	O
efficient	O
training	O
algorithms	O
than	O
are	O
available	O
for	O
the	O
general	O
class	O
of	O
Boltzmann	B-Algorithm
machines	I-Algorithm
,	O
in	O
particular	O
the	O
gradient-based	B-Algorithm
contrastive	B-Algorithm
divergence	I-Algorithm
algorithm	O
.	O
</s>
<s>
Restricted	B-Algorithm
Boltzmann	I-Algorithm
machines	I-Algorithm
can	O
also	O
be	O
used	O
in	O
deep	B-Algorithm
learning	I-Algorithm
networks	O
.	O
</s>
<s>
In	O
particular	O
,	O
deep	B-Algorithm
belief	I-Algorithm
networks	I-Algorithm
can	O
be	O
formed	O
by	O
"	O
stacking	O
"	O
RBMs	O
and	O
optionally	O
fine-tuning	O
the	O
resulting	O
deep	O
network	O
with	O
gradient	B-Algorithm
descent	I-Algorithm
and	O
backpropagation	B-Algorithm
.	O
</s>
<s>
The	O
standard	O
type	O
of	O
RBM	O
has	O
binary-valued	O
(	O
Boolean	O
)	O
hidden	O
and	O
visible	O
units	O
,	O
and	O
consists	O
of	O
a	O
matrix	B-Architecture
of	O
weights	O
of	O
size	O
.	O
</s>
<s>
Each	O
weight	O
element	O
of	O
the	O
matrix	B-Architecture
is	O
associated	O
with	O
the	O
connection	O
between	O
the	O
visible	O
(	O
input	O
)	O
unit	O
and	O
the	O
hidden	O
unit	O
.	O
</s>
<s>
or	O
,	O
in	O
matrix	B-Architecture
notation	O
,	O
</s>
<s>
This	O
energy	O
function	O
is	O
analogous	O
to	O
that	O
of	O
a	O
Hopfield	B-Algorithm
network	I-Algorithm
.	O
</s>
<s>
As	O
with	O
general	O
Boltzmann	B-Algorithm
machines	I-Algorithm
,	O
the	O
joint	O
probability	O
distribution	O
for	O
the	O
visible	O
and	O
hidden	O
vectors	O
is	O
defined	O
in	O
terms	O
of	O
the	O
energy	O
function	O
as	O
follows	O
,	O
</s>
<s>
The	O
visible	O
units	O
of	O
Restricted	B-Algorithm
Boltzmann	I-Algorithm
Machine	I-Algorithm
can	O
be	O
multinomial	O
,	O
although	O
the	O
hidden	O
units	O
are	O
Bernoulli	O
.	O
</s>
<s>
They	O
are	O
applied	O
in	O
topic	O
modeling	O
,	O
and	O
recommender	B-Application
systems	I-Application
.	O
</s>
<s>
Restricted	B-Algorithm
Boltzmann	I-Algorithm
machines	I-Algorithm
are	O
a	O
special	O
case	O
of	O
Boltzmann	B-Algorithm
machines	I-Algorithm
and	O
Markov	O
random	O
fields	O
.	O
</s>
<s>
Restricted	B-Algorithm
Boltzmann	I-Algorithm
machines	I-Algorithm
are	O
trained	O
to	O
maximize	O
the	O
product	O
of	O
probabilities	O
assigned	O
to	O
some	O
training	O
set	O
(	O
a	O
matrix	B-Architecture
,	O
each	O
row	O
of	O
which	O
is	O
treated	O
as	O
a	O
visible	O
vector	O
)	O
,	O
</s>
<s>
The	O
algorithm	O
most	O
often	O
used	O
to	O
train	O
RBMs	O
,	O
that	O
is	O
,	O
to	O
optimize	O
the	O
weight	O
matrix	B-Architecture
,	O
is	O
the	O
contrastive	B-Algorithm
divergence	I-Algorithm
(	O
CD	O
)	O
algorithm	O
due	O
to	O
Hinton	O
,	O
originally	O
developed	O
to	O
train	O
PoE	O
(	O
product	B-General_Concept
of	I-General_Concept
experts	I-General_Concept
)	O
models	O
.	O
</s>
<s>
The	O
algorithm	O
performs	O
Gibbs	B-Algorithm
sampling	I-Algorithm
and	O
is	O
used	O
inside	O
a	O
gradient	B-Algorithm
descent	I-Algorithm
procedure	O
(	O
similar	O
to	O
the	O
way	O
backpropagation	B-Algorithm
is	O
used	O
inside	O
such	O
a	O
procedure	O
when	O
training	O
feedforward	O
neural	B-Architecture
nets	I-Architecture
)	O
to	O
compute	O
weight	O
update	O
.	O
</s>
<s>
The	O
basic	O
,	O
single-step	O
contrastive	B-Algorithm
divergence	I-Algorithm
(	O
CD-1	O
)	O
procedure	O
for	O
a	O
single	O
sample	O
can	O
be	O
summarized	O
as	O
follows	O
:	O
</s>
<s>
Let	O
the	O
update	O
to	O
the	O
weight	O
matrix	B-Architecture
be	O
the	O
positive	O
gradient	O
minus	O
the	O
negative	O
gradient	O
,	O
times	O
some	O
learning	O
rate	O
:	O
.	O
</s>
<s>
The	O
difference	O
between	O
the	O
Stacked	O
Restricted	B-Algorithm
Boltzmann	I-Algorithm
Machines	I-Algorithm
and	O
RBM	O
is	O
that	O
RBM	O
has	O
lateral	O
connections	O
within	O
a	O
layer	O
that	O
are	O
prohibited	O
to	O
make	O
analysis	O
tractable	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
Stacked	O
Boltzmann	O
consists	O
of	O
a	O
combination	O
of	O
an	O
unsupervised	B-General_Concept
three-layer	O
network	O
with	O
symmetric	O
weights	O
and	O
a	O
supervised	B-General_Concept
fine-tuned	O
top	O
layer	O
for	O
recognizing	O
three	O
classes	O
.	O
</s>
<s>
The	O
usage	O
of	O
Stacked	O
Boltzmann	O
is	O
to	O
understand	O
Natural	O
languages	O
,	O
retrieve	O
documents	O
,	O
image	O
generation	O
,	O
and	O
classification	B-General_Concept
.	O
</s>
<s>
These	O
functions	O
are	O
trained	O
with	O
unsupervised	B-General_Concept
pre-training	O
and/or	O
supervised	B-General_Concept
fine-tuning	O
.	O
</s>
<s>
Stacked	O
Boltzmann	O
does	O
share	O
similarities	O
with	O
RBM	O
,	O
the	O
neuron	O
for	O
Stacked	O
Boltzmann	O
is	O
a	O
stochastic	O
binary	O
Hopfield	O
neuron	O
,	O
which	O
is	O
the	O
same	O
as	O
the	O
Restricted	B-Algorithm
Boltzmann	I-Algorithm
Machine	I-Algorithm
.	O
</s>
<s>
Restricted	O
Boltzmann	O
train	O
one	O
layer	O
at	O
a	O
time	O
and	O
approximate	O
equilibrium	O
state	O
with	O
a	O
3-segment	O
pass	O
,	O
not	O
performing	O
back	B-Algorithm
propagation	I-Algorithm
.	O
</s>
<s>
Restricted	O
Boltzmann	O
uses	O
both	O
supervised	B-General_Concept
and	O
unsupervised	B-General_Concept
on	O
different	O
RBM	O
for	O
pre-training	O
for	O
classification	B-General_Concept
and	O
recognition	O
.	O
</s>
<s>
It	O
does	O
not	O
follow	O
the	O
gradient	O
of	O
any	O
function	O
,	O
so	O
the	O
approximation	O
of	O
Contrastive	B-Algorithm
divergence	I-Algorithm
to	O
maximum	O
likelihood	O
is	O
improvised	O
.	O
</s>
