<s>
In	O
signal	O
processing	O
,	O
independent	B-Algorithm
component	I-Algorithm
analysis	I-Algorithm
(	O
ICA	O
)	O
is	O
a	O
computational	O
method	O
for	O
separating	O
a	O
multivariate	B-General_Concept
signal	O
into	O
additive	O
subcomponents	O
.	O
</s>
<s>
ICA	O
is	O
a	O
special	O
case	O
of	O
blind	B-Application
source	I-Application
separation	I-Application
.	O
</s>
<s>
Independent	B-Algorithm
component	I-Algorithm
analysis	I-Algorithm
attempts	O
to	O
decompose	O
a	O
multivariate	B-General_Concept
signal	O
into	O
independent	O
non-Gaussian	O
signals	O
.	O
</s>
<s>
The	O
Minimization-of-Mutual	O
information	O
(	O
MMI	O
)	O
family	O
of	O
ICA	O
algorithms	O
uses	O
measures	O
like	O
Kullback-Leibler	O
Divergence	O
and	O
maximum	O
entropy	B-Algorithm
.	O
</s>
<s>
The	O
non-Gaussianity	O
family	O
of	O
ICA	O
algorithms	O
,	O
motivated	O
by	O
the	O
central	O
limit	O
theorem	O
,	O
uses	O
kurtosis	B-Error_Name
and	O
negentropy	O
.	O
</s>
<s>
Typical	O
algorithms	O
for	O
ICA	O
use	O
centering	O
(	O
subtract	O
the	O
mean	O
to	O
create	O
a	O
zero	O
mean	O
signal	O
)	O
,	O
whitening	B-Algorithm
(	O
usually	O
with	O
the	O
eigenvalue	O
decomposition	O
)	O
,	O
and	O
dimensionality	B-Algorithm
reduction	I-Algorithm
as	O
preprocessing	O
steps	O
in	O
order	O
to	O
simplify	O
and	O
reduce	O
the	O
complexity	O
of	O
the	O
problem	O
for	O
the	O
actual	O
iterative	O
algorithm	O
.	O
</s>
<s>
Whitening	B-Algorithm
and	O
dimension	B-Algorithm
reduction	I-Algorithm
can	O
be	O
achieved	O
with	O
principal	B-Application
component	I-Application
analysis	I-Application
or	O
singular	O
value	O
decomposition	O
.	O
</s>
<s>
Whitening	B-Algorithm
ensures	O
that	O
all	O
dimensions	O
are	O
treated	O
equally	O
a	O
priori	O
before	O
the	O
algorithm	O
is	O
run	O
.	O
</s>
<s>
Well-known	O
algorithms	O
for	O
ICA	O
include	O
infomax	O
,	O
FastICA	B-Algorithm
,	O
JADE	B-Algorithm
,	O
and	O
kernel-independent	B-Algorithm
component	I-Algorithm
analysis	I-Algorithm
,	O
among	O
others	O
.	O
</s>
<s>
ICA	O
is	O
important	O
to	O
blind	B-Application
signal	I-Application
separation	I-Application
and	O
has	O
many	O
practical	O
applications	O
.	O
</s>
<s>
It	O
is	O
closely	O
related	O
to	O
(	O
or	O
even	O
a	O
special	O
case	O
of	O
)	O
the	O
search	O
for	O
a	O
factorial	B-Algorithm
code	I-Algorithm
of	O
the	O
data	O
,	O
i.e.	O
,	O
a	O
new	O
vector-valued	O
representation	O
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>
Linear	O
independent	B-Algorithm
component	I-Algorithm
analysis	I-Algorithm
can	O
be	O
divided	O
into	O
noiseless	O
and	O
noisy	O
cases	O
,	O
where	O
noiseless	O
ICA	O
is	O
a	O
special	O
case	O
of	O
noisy	O
ICA	O
.	O
</s>
<s>
The	O
data	O
are	O
represented	O
by	O
the	O
observed	O
random	B-General_Concept
vector	I-General_Concept
and	O
the	O
hidden	O
components	O
as	O
the	O
random	B-General_Concept
vector	I-General_Concept
The	O
task	O
is	O
to	O
transform	O
the	O
observed	O
data	O
using	O
a	O
linear	O
static	O
transformation	O
as	O
into	O
a	O
vector	O
of	O
maximally	O
independent	O
components	O
measured	O
by	O
some	O
function	O
of	O
independence	O
.	O
</s>
<s>
The	O
components	O
of	O
the	O
observed	O
random	B-General_Concept
vector	I-General_Concept
are	O
generated	O
as	O
a	O
sum	O
of	O
the	O
independent	O
components	O
,	O
:	O
</s>
<s>
The	O
same	O
generative	O
model	O
can	O
be	O
written	O
in	O
vector	O
form	O
as	O
,	O
where	O
the	O
observed	O
random	B-General_Concept
vector	I-General_Concept
is	O
represented	O
by	O
the	O
basis	O
vectors	O
.	O
</s>
<s>
Given	O
the	O
model	O
and	O
realizations	O
(	O
samples	O
)	O
of	O
the	O
random	B-General_Concept
vector	I-General_Concept
,	O
the	O
task	O
is	O
to	O
estimate	O
both	O
the	O
mixing	O
matrix	O
and	O
the	O
sources	O
.	O
</s>
<s>
If	O
the	O
number	O
of	O
basis	O
vectors	O
is	O
greater	O
than	O
the	O
dimensionality	O
of	O
the	O
observed	O
vectors	O
,	O
,	O
the	O
task	O
is	O
overcomplete	O
but	O
is	O
still	O
solvable	O
with	O
the	O
pseudo	B-Algorithm
inverse	I-Algorithm
.	O
</s>
<s>
The	O
above	O
problem	O
can	O
be	O
heuristically	O
solved	O
by	O
assuming	O
variables	O
are	O
continuous	O
and	O
running	O
FastICA	B-Algorithm
on	O
binary	O
observation	O
data	O
to	O
get	O
the	O
mixing	O
matrix	O
(	O
real	O
values	O
)	O
,	O
then	O
apply	O
round	O
number	O
techniques	O
on	O
to	O
obtain	O
the	O
binary	O
values	O
.	O
</s>
<s>
Another	O
method	O
is	O
to	O
use	O
dynamic	B-Algorithm
programming	I-Algorithm
:	O
recursively	O
breaking	O
the	O
observation	O
matrix	O
into	O
its	O
sub-matrices	O
and	O
run	O
the	O
inference	O
algorithm	O
on	O
these	O
sub-matrices	O
.	O
</s>
<s>
Although	O
this	O
problem	O
appears	O
quite	O
complex	O
,	O
it	O
can	O
be	O
accurately	O
solved	O
with	O
a	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
search	O
tree	O
algorithm	O
or	O
tightly	O
upper	O
bounded	O
with	O
a	O
single	O
multiplication	O
of	O
a	O
matrix	O
with	O
a	O
vector	O
.	O
</s>
<s>
One	O
type	O
of	O
method	O
for	O
doing	O
so	O
is	O
projection	B-General_Concept
pursuit	I-General_Concept
.	O
</s>
<s>
Projection	B-General_Concept
pursuit	I-General_Concept
seeks	O
one	O
projection	O
at	O
a	O
time	O
such	O
that	O
the	O
extracted	O
signal	O
is	O
as	O
non-Gaussian	O
as	O
possible	O
.	O
</s>
<s>
One	O
practical	O
advantage	O
of	O
projection	B-General_Concept
pursuit	I-General_Concept
over	O
ICA	O
is	O
that	O
fewer	O
than	O
M	O
signals	O
can	O
be	O
extracted	O
if	O
required	O
,	O
where	O
each	O
source	O
signal	O
is	O
extracted	O
from	O
M	O
signal	O
mixtures	O
using	O
an	O
M-element	O
weight	O
vector	O
.	O
</s>
<s>
We	O
can	O
use	O
kurtosis	B-Error_Name
to	O
recover	O
the	O
multiple	O
source	O
signal	O
by	O
finding	O
the	O
correct	O
weight	O
vectors	O
with	O
the	O
use	O
of	O
projection	B-General_Concept
pursuit	I-General_Concept
.	O
</s>
<s>
The	O
constant	O
3	O
ensures	O
that	O
Gaussian	O
signals	O
have	O
zero	O
kurtosis	B-Error_Name
,	O
Super-Gaussian	O
signals	O
have	O
positive	O
kurtosis	B-Error_Name
,	O
and	O
Sub-Gaussian	O
signals	O
have	O
negative	O
kurtosis	B-Error_Name
.	O
</s>
<s>
The	O
denominator	O
is	O
the	O
variance	O
of	O
,	O
and	O
ensures	O
that	O
the	O
measured	O
kurtosis	B-Error_Name
takes	O
account	O
of	O
signal	O
variance	O
.	O
</s>
<s>
The	O
goal	O
of	O
projection	B-General_Concept
pursuit	I-General_Concept
is	O
to	O
maximize	O
the	O
kurtosis	B-Error_Name
,	O
and	O
make	O
the	O
extracted	O
signal	O
as	O
non-normal	O
as	O
possible	O
.	O
</s>
<s>
Using	O
kurtosis	B-Error_Name
as	O
a	O
measure	O
of	O
non-normality	O
,	O
we	O
can	O
now	O
examine	O
how	O
the	O
kurtosis	B-Error_Name
of	O
a	O
signal	O
extracted	O
from	O
a	O
set	O
of	O
M	O
mixtures	O
varies	O
as	O
the	O
weight	O
vector	O
is	O
rotated	O
around	O
the	O
origin	O
.	O
</s>
<s>
the	O
kurtosis	B-Error_Name
of	O
the	O
extracted	O
signal	O
to	O
be	O
maximal	O
precisely	O
when	O
.	O
</s>
<s>
the	O
kurtosis	B-Error_Name
of	O
the	O
extracted	O
signal	O
to	O
be	O
maximal	O
when	O
is	O
orthogonal	O
to	O
the	O
projected	O
axes	O
or	O
,	O
because	O
we	O
know	O
the	O
optimal	O
weight	O
vector	O
should	O
be	O
orthogonal	O
to	O
a	O
transformed	O
axis	O
or	O
.	O
</s>
<s>
For	O
multiple	O
source	O
mixture	O
signals	O
,	O
we	O
can	O
use	O
kurtosis	B-Error_Name
and	O
Gram-Schmidt	B-Algorithm
Orthogonalization	I-Algorithm
(	O
GSO	O
)	O
to	O
recover	O
the	O
signals	O
.	O
</s>
<s>
In	O
order	O
to	O
find	O
the	O
correct	O
value	O
of	O
,	O
we	O
can	O
use	O
gradient	B-Algorithm
descent	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
kurtosis	B-Error_Name
can	O
thus	O
be	O
written	O
as	O
:	O
</s>
<s>
Another	O
approach	O
is	O
using	O
negentropy	O
instead	O
of	O
kurtosis	B-Error_Name
.	O
</s>
<s>
Using	O
negentropy	O
is	O
a	O
more	O
robust	O
method	O
than	O
kurtosis	B-Error_Name
,	O
as	O
kurtosis	B-Error_Name
is	O
very	O
sensitive	O
to	O
outliers	O
.	O
</s>
<s>
The	O
negentropy	O
methods	O
are	O
based	O
on	O
an	O
important	O
property	O
of	O
Gaussian	O
distribution	O
:	O
a	O
Gaussian	O
variable	O
has	O
the	O
largest	O
entropy	B-Algorithm
among	O
all	O
continuous	O
random	O
variables	O
of	O
equal	O
variance	O
.	O
</s>
<s>
A	O
simple	O
proof	O
can	O
be	O
found	O
in	O
Differential	O
entropy	B-Algorithm
.	O
</s>
<s>
A	O
proof	O
can	O
be	O
found	O
in	O
the	O
original	O
papers	O
of	O
Comon	O
;	O
it	O
has	O
been	O
reproduced	O
in	O
the	O
book	O
Independent	B-Algorithm
Component	I-Algorithm
Analysis	I-Algorithm
by	O
Aapo	O
Hyvärinen	O
,	O
Juha	O
Karhunen	O
,	O
and	O
Erkki	O
Oja	O
This	O
approximation	O
also	O
suffers	O
from	O
the	O
same	O
problem	O
as	O
kurtosis	B-Error_Name
(	O
sensitivity	O
to	O
outliers	O
)	O
.	O
</s>
<s>
Infomax	O
ICA	O
is	O
essentially	O
a	O
multivariate	B-General_Concept
,	O
parallel	O
version	O
of	O
projection	B-General_Concept
pursuit	I-General_Concept
.	O
</s>
<s>
Whereas	O
projection	B-General_Concept
pursuit	I-General_Concept
extracts	O
a	O
series	O
of	O
signals	O
one	O
at	O
a	O
time	O
from	O
a	O
set	O
of	O
M	O
signal	O
mixtures	O
,	O
ICA	O
extracts	O
M	O
signals	O
in	O
parallel	O
.	O
</s>
<s>
This	O
tends	O
to	O
make	O
ICA	O
more	O
robust	O
than	O
projection	B-General_Concept
pursuit	I-General_Concept
.	O
</s>
<s>
The	O
projection	B-General_Concept
pursuit	I-General_Concept
method	O
uses	O
Gram-Schmidt	B-Algorithm
orthogonalization	I-Algorithm
to	O
ensure	O
the	O
independence	O
of	O
the	O
extracted	O
signal	O
,	O
while	O
ICA	O
use	O
infomax	O
and	O
maximum	O
likelihood	O
estimate	O
to	O
ensure	O
the	O
independence	O
of	O
the	O
extracted	O
signal	O
.	O
</s>
<s>
The	O
process	O
of	O
ICA	O
based	O
on	O
infomax	O
in	O
short	O
is	O
:	O
given	O
a	O
set	O
of	O
signal	O
mixtures	O
and	O
a	O
set	O
of	O
identical	O
independent	O
model	O
cumulative	O
distribution	O
functions(cdfs )	O
,	O
we	O
seek	O
the	O
unmixing	O
matrix	O
which	O
maximizes	O
the	O
joint	O
entropy	B-Algorithm
of	O
the	O
signals	O
,	O
where	O
are	O
the	O
signals	O
extracted	O
by	O
.	O
</s>
<s>
Given	O
the	O
optimal	O
,	O
the	O
signals	O
have	O
maximum	O
entropy	B-Algorithm
and	O
are	O
therefore	O
independent	O
,	O
which	O
ensures	O
that	O
the	O
extracted	O
signals	O
are	O
also	O
independent	O
.	O
</s>
<s>
Note	O
that	O
if	O
the	O
source	O
signal	O
model	O
probability	O
density	O
function	O
matches	O
the	O
probability	O
density	O
function	O
of	O
the	O
extracted	O
signal	O
,	O
then	O
maximizing	O
the	O
joint	O
entropy	B-Algorithm
of	O
also	O
maximizes	O
the	O
amount	O
of	O
mutual	O
information	O
between	O
and	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
using	O
entropy	B-Algorithm
to	O
extract	O
independent	O
signals	O
is	O
known	O
as	O
infomax	O
.	O
</s>
<s>
Consider	O
the	O
entropy	B-Algorithm
of	O
the	O
vector	O
variable	O
,	O
where	O
is	O
the	O
set	O
of	O
signals	O
extracted	O
by	O
the	O
unmixing	O
matrix	O
.	O
</s>
<s>
For	O
a	O
finite	O
set	O
of	O
values	O
sampled	O
from	O
a	O
distribution	O
with	O
pdf	O
,	O
the	O
entropy	B-Algorithm
of	O
can	O
be	O
estimated	O
as	O
:	O
</s>
<s>
The	O
joint	O
pdf	O
can	O
be	O
shown	O
to	O
be	O
related	O
to	O
the	O
joint	O
pdf	O
of	O
the	O
extracted	O
signals	O
by	O
the	O
multivariate	B-General_Concept
form	O
:	O
</s>
<s>
Like	O
the	O
projection	B-General_Concept
pursuit	I-General_Concept
situation	O
,	O
we	O
can	O
use	O
gradient	B-Algorithm
descent	I-Algorithm
method	I-Algorithm
to	O
find	O
the	O
optimal	O
solution	O
of	O
the	O
unmixing	O
matrix	O
.	O
</s>
<s>
The	O
early	O
general	O
framework	O
for	O
independent	B-Algorithm
component	I-Algorithm
analysis	I-Algorithm
was	O
introduced	O
by	O
Jeanny	O
Hérault	O
and	O
Bernard	O
Ans	O
from	O
1984	O
,	O
further	O
developed	O
by	O
Christian	O
Jutten	O
in	O
1985	O
and	O
1986	O
,	O
and	O
refined	O
by	O
Pierre	O
Comon	O
in	O
1991	O
,	O
and	O
popularized	O
in	O
his	O
paper	O
of	O
1994	O
.	O
</s>
<s>
A	O
largely	O
used	O
one	O
,	O
including	O
in	O
industrial	O
applications	O
,	O
is	O
the	O
FastICA	B-Algorithm
algorithm	O
,	O
developed	O
by	O
Hyvärinen	O
and	O
Oja	O
,	O
which	O
uses	O
the	O
negentropy	O
as	O
cost	O
function	O
.	O
</s>
<s>
Other	O
examples	O
are	O
rather	O
related	O
to	O
blind	B-Application
source	I-Application
separation	I-Application
where	O
a	O
more	O
general	O
approach	O
is	O
used	O
.	O
</s>
<s>
removing	O
artifacts	O
,	O
such	O
as	O
eye	O
blinks	O
,	O
from	O
EEG	B-Application
data	O
.	O
</s>
<s>
studies	O
of	O
the	O
resting	B-Algorithm
state	I-Algorithm
network	I-Algorithm
of	O
the	O
brain	O
.	O
</s>
