<s>
is	O
an	O
extension	O
of	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
using	O
techniques	O
of	O
kernel	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
To	O
understand	O
the	O
utility	O
of	O
kernel	B-Algorithm
PCA	I-Algorithm
,	O
particularly	O
for	O
clustering	O
,	O
observe	O
that	O
,	O
while	O
N	O
points	O
cannot	O
,	O
in	O
general	O
,	O
be	O
linearly	O
separated	O
in	O
dimensions	O
,	O
they	O
can	O
almost	O
always	O
be	O
linearly	O
separated	O
in	O
dimensions	O
.	O
</s>
<s>
Instead	O
,	O
in	O
kernel	B-Algorithm
PCA	I-Algorithm
,	O
a	O
non-trivial	O
,	O
arbitrary	O
function	O
is	O
'	O
chosen	O
 '	O
that	O
is	O
never	O
calculated	O
explicitly	O
,	O
allowing	O
the	O
possibility	O
to	O
use	O
very-high-dimensional	O
'	O
s	O
if	O
we	O
never	O
have	O
to	O
actually	O
evaluate	O
the	O
data	O
in	O
that	O
space	O
.	O
</s>
<s>
which	O
represents	O
the	O
inner	O
product	O
space	O
(	O
see	O
Gramian	B-Algorithm
matrix	I-Algorithm
)	O
of	O
the	O
otherwise	O
intractable	O
feature	O
space	O
.	O
</s>
<s>
Because	O
we	O
are	O
never	O
working	O
directly	O
in	O
the	O
feature	O
space	O
,	O
the	O
kernel-formulation	O
of	O
PCA	O
is	O
restricted	O
in	O
that	O
it	O
computes	O
not	O
the	O
principal	B-Application
components	I-Application
themselves	O
,	O
but	O
the	O
projections	O
of	O
our	O
data	O
onto	O
those	O
components	O
.	O
</s>
<s>
We	O
use	O
to	O
perform	O
the	O
kernel	B-Algorithm
PCA	I-Algorithm
algorithm	O
described	O
above	O
.	O
</s>
<s>
One	O
caveat	O
of	O
kernel	B-Algorithm
PCA	I-Algorithm
should	O
be	O
illustrated	O
here	O
.	O
</s>
<s>
In	O
linear	O
PCA	O
,	O
we	O
can	O
use	O
the	O
eigenvalues	O
to	O
rank	O
the	O
eigenvectors	O
based	O
on	O
how	O
much	O
of	O
the	O
variation	O
of	O
the	O
data	O
is	O
captured	O
by	O
each	O
principal	B-Application
component	I-Application
.	O
</s>
<s>
Consider	O
three	O
concentric	O
clouds	O
of	O
points	O
(	O
shown	O
)	O
;	O
we	O
wish	O
to	O
use	O
kernel	B-Algorithm
PCA	I-Algorithm
to	O
identify	O
these	O
groups	O
.	O
</s>
<s>
Applying	O
this	O
to	O
kernel	B-Algorithm
PCA	I-Algorithm
yields	O
the	O
next	O
image	O
.	O
</s>
<s>
Now	O
consider	O
a	O
Gaussian	B-Application
kernel	O
:	O
</s>
<s>
Note	O
in	O
particular	O
that	O
the	O
first	O
principal	B-Application
component	I-Application
is	O
enough	O
to	O
distinguish	O
the	O
three	O
different	O
groups	O
,	O
which	O
is	O
impossible	O
using	O
only	O
linear	O
PCA	O
,	O
because	O
linear	O
PCA	O
operates	O
only	O
in	O
the	O
given	O
(	O
in	O
this	O
case	O
two-dimensional	O
)	O
space	O
,	O
in	O
which	O
these	O
concentric	O
point	O
clouds	O
are	O
not	O
linearly	O
separable	O
.	O
</s>
<s>
Kernel	B-Algorithm
PCA	I-Algorithm
has	O
been	O
demonstrated	O
to	O
be	O
useful	O
for	O
novelty	O
detection	O
and	O
image	O
de-noising	O
.	O
</s>
