<s>
When	O
this	O
occurs	O
,	O
it	O
can	O
be	O
desirable	O
to	O
create	O
a	O
factorial	B-Algorithm
code	I-Algorithm
of	O
the	O
data	O
,	O
i	O
.	O
e.	O
,	O
a	O
new	O
vector-valued	O
representation	B-Algorithm
of	O
each	O
data	O
vector	O
such	O
that	O
it	O
gets	O
uniquely	O
encoded	O
by	O
the	O
resulting	O
code	O
vector	O
(	O
loss-free	O
coding	O
)	O
,	O
but	O
the	O
code	O
components	O
are	O
statistically	O
independent	O
.	O
</s>
<s>
Later	O
supervised	B-General_Concept
learning	I-General_Concept
usually	O
works	O
much	O
better	O
when	O
the	O
raw	O
input	O
data	O
is	O
first	O
translated	O
into	O
such	O
a	O
factorial	B-Algorithm
code	I-Algorithm
.	O
</s>
<s>
A	O
naive	B-General_Concept
Bayes	I-General_Concept
classifier	I-General_Concept
will	O
assume	O
the	O
pixels	O
are	O
statistically	O
independent	O
random	O
variables	O
and	O
therefore	O
fail	O
to	O
produce	O
good	O
results	O
.	O
</s>
<s>
If	O
the	O
data	O
are	O
first	O
encoded	O
in	O
a	O
factorial	O
way	O
,	O
however	O
,	O
then	O
the	O
naive	B-General_Concept
Bayes	I-General_Concept
classifier	I-General_Concept
will	O
achieve	O
its	O
optimal	O
performance	O
(	O
compare	O
Schmidhuber	O
et	O
al	O
.	O
</s>
<s>
To	O
create	O
factorial	B-Algorithm
codes	I-Algorithm
,	O
Horace	O
Barlow	O
and	O
co-workers	O
suggested	O
to	O
minimize	O
the	O
sum	O
of	O
the	O
bit	O
entropies	O
of	O
the	O
code	O
components	O
of	O
binary	O
codes	O
(	O
1989	O
)	O
.	O
</s>
<s>
Jürgen	O
Schmidhuber	O
(	O
1992	O
)	O
re-formulated	O
the	O
problem	O
in	O
terms	O
of	O
predictors	O
and	O
binary	O
feature	B-Algorithm
detectors	B-Application
,	O
each	O
receiving	O
the	O
raw	O
data	O
as	O
an	O
input	O
.	O
</s>
<s>
For	O
each	O
detector	B-Application
there	O
is	O
a	O
predictor	O
that	O
sees	O
the	O
other	O
detectors	B-Application
and	O
learns	O
to	O
predict	O
the	O
output	O
of	O
its	O
own	O
detector	B-Application
in	O
response	O
to	O
the	O
various	O
input	O
vectors	O
or	O
images	O
.	O
</s>
<s>
But	O
each	O
detector	B-Application
uses	O
a	O
machine	O
learning	O
algorithm	O
to	O
become	O
as	O
unpredictable	O
as	O
possible	O
.	O
</s>
<s>
The	O
global	O
optimum	O
of	O
this	O
objective	O
function	O
corresponds	O
to	O
a	O
factorial	B-Algorithm
code	I-Algorithm
represented	O
in	O
a	O
distributed	O
fashion	O
across	O
the	O
outputs	O
of	O
the	O
feature	B-Algorithm
detectors	B-Application
.	O
</s>
<s>
Painsky	O
,	O
Rosset	O
and	O
Feder	O
(	O
2016	O
,	O
2017	O
)	O
further	O
studied	O
this	O
problem	O
in	O
the	O
context	O
of	O
independent	B-Algorithm
component	I-Algorithm
analysis	I-Algorithm
over	O
finite	O
alphabet	O
sizes	O
.	O
</s>
<s>
In	O
addition	O
,	O
they	O
demonstrate	O
the	O
use	O
of	O
factorial	B-Algorithm
codes	I-Algorithm
to	O
data	O
compression	O
in	O
multiple	O
setups	O
(	O
2017	O
)	O
.	O
</s>
