<s>
Oja	O
's	O
learning	B-Algorithm
rule	I-Algorithm
,	O
or	O
simply	O
Oja	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
,	O
named	O
after	O
Finnish	O
computer	O
scientist	O
Erkki	O
Oja	O
,	O
is	O
a	O
model	O
of	O
how	O
neurons	O
in	O
the	O
brain	O
or	O
in	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
change	O
connection	O
strength	O
,	O
or	O
learn	O
,	O
over	O
time	O
.	O
</s>
<s>
It	O
is	O
a	O
modification	O
of	O
the	O
standard	O
Hebb	O
's	O
Rule	O
(	O
see	O
Hebbian	O
learning	O
)	O
that	O
,	O
through	O
multiplicative	O
normalization	O
,	O
solves	O
all	O
stability	O
problems	O
and	O
generates	O
an	O
algorithm	O
for	O
principal	B-Application
components	I-Application
analysis	I-Application
.	O
</s>
<s>
Oja	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
requires	O
a	O
number	O
of	O
simplifications	O
to	O
derive	O
,	O
but	O
in	O
its	O
final	O
form	O
it	O
is	O
demonstrably	O
stable	O
,	O
unlike	O
Hebb	O
's	O
rule	O
.	O
</s>
<s>
It	O
is	O
a	O
single-neuron	O
special	O
case	O
of	O
the	O
Generalized	B-Algorithm
Hebbian	I-Algorithm
Algorithm	I-Algorithm
.	O
</s>
<s>
However	O
,	O
Oja	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
can	O
also	O
be	O
generalized	O
in	O
other	O
ways	O
to	O
varying	O
degrees	O
of	O
stability	O
and	O
success	O
.	O
</s>
<s>
The	O
simplest	O
learning	B-Algorithm
rule	I-Algorithm
known	O
is	O
Hebb	O
's	O
rule	O
,	O
which	O
states	O
in	O
conceptual	O
terms	O
that	O
neurons	O
that	O
fire	O
together	O
,	O
wire	O
together	O
.	O
</s>
<s>
In	O
analyzing	O
the	O
convergence	O
of	O
a	O
single	O
neuron	O
evolving	O
by	O
Oja	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
,	O
one	O
extracts	O
the	O
first	O
principal	B-Application
component	I-Application
,	O
or	O
feature	O
,	O
of	O
a	O
data	O
set	O
.	O
</s>
<s>
Furthermore	O
,	O
with	O
extensions	O
using	O
the	O
Generalized	B-Algorithm
Hebbian	I-Algorithm
Algorithm	I-Algorithm
,	O
one	O
can	O
create	O
a	O
multi-Oja	O
neural	B-Architecture
network	I-Architecture
that	O
can	O
extract	O
as	O
many	O
features	O
as	O
desired	O
,	O
allowing	O
for	O
principal	B-Application
components	I-Application
analysis	I-Application
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
single	O
neuron	O
trained	O
by	O
Oja	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
,	O
we	O
find	O
the	O
weight	O
vector	O
converges	O
to	O
,	O
or	O
the	O
first	O
principal	B-Application
component	I-Application
,	O
as	O
time	O
or	O
number	O
of	O
iterations	O
approaches	O
infinity	O
.	O
</s>
<s>
These	O
results	O
are	O
derived	O
using	O
Lyapunov	O
function	O
analysis	O
,	O
and	O
they	O
show	O
that	O
Oja	O
's	O
neuron	O
necessarily	O
converges	O
on	O
strictly	O
the	O
first	O
principal	B-Application
component	I-Application
if	O
certain	O
conditions	O
are	O
met	O
in	O
our	O
original	O
learning	B-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
Our	O
output	O
activation	B-Algorithm
function	I-Algorithm
is	O
also	O
allowed	O
to	O
be	O
nonlinear	O
and	O
nonstatic	O
,	O
but	O
it	O
must	O
be	O
continuously	O
differentiable	O
in	O
both	O
and	O
and	O
have	O
derivatives	O
bounded	O
in	O
time	O
.	O
</s>
<s>
Recently	O
,	O
in	O
the	O
context	O
of	O
associative	O
learning	O
,	O
it	O
has	O
been	O
shown	O
that	O
the	O
Hebbian	O
rule	O
,	O
which	O
is	O
similar	O
to	O
Oja	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
,	O
can	O
be	O
generalized	O
using	O
an	O
Ising-like	O
model	O
:	O
The	O
main	O
idea	O
of	O
the	O
generalization	O
is	O
based	O
on	O
formulating	O
the	O
energy	O
function	O
like	O
in	O
Ising	O
model	O
and	O
then	O
applying	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
algorithm	O
to	O
it	O
.	O
</s>
<s>
Oja	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
was	O
originally	O
described	O
in	O
Oja	O
's	O
1982	O
paper	O
,	O
but	O
the	O
principle	O
of	O
self-organization	O
to	O
which	O
it	O
is	O
applied	O
is	O
first	O
attributed	O
to	O
Alan	O
Turing	O
in	O
1952	O
.	O
</s>
<s>
PCA	O
has	O
also	O
had	O
a	O
long	O
history	O
of	O
use	O
before	O
Oja	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
formalized	O
its	O
use	O
in	O
network	O
computation	O
in	O
1989	O
.	O
</s>
<s>
Therefore	O
,	O
Oja	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
has	O
an	O
important	O
place	O
in	O
image	O
and	O
speech	O
processing	O
.	O
</s>
<s>
A	O
canonical	O
example	O
is	O
its	O
use	O
in	O
binocular	B-Algorithm
vision	I-Algorithm
.	O
</s>
<s>
There	O
is	O
clear	O
evidence	O
for	O
both	O
long-term	O
potentiation	O
and	O
long-term	O
depression	O
in	O
biological	O
neural	B-Architecture
networks	I-Architecture
,	O
along	O
with	O
a	O
normalization	O
effect	O
in	O
both	O
input	O
weights	O
and	O
neuron	O
outputs	O
.	O
</s>
<s>
However	O
,	O
while	O
there	O
is	O
no	O
direct	O
experimental	O
evidence	O
yet	O
of	O
Oja	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
active	O
in	O
a	O
biological	O
neural	B-Architecture
network	I-Architecture
,	O
a	O
biophysical	O
derivation	O
of	O
a	O
generalization	O
of	O
the	O
rule	O
is	O
possible	O
.	O
</s>
<s>
Note	O
that	O
the	O
angle	O
brackets	O
denote	O
the	O
average	O
and	O
the	O
∗	O
operator	O
is	O
a	O
convolution	B-Language
.	O
</s>
