<s>
In	O
machine	O
learning	O
,	O
kernel	B-Algorithm
machines	I-Algorithm
are	O
a	O
class	O
of	O
algorithms	O
for	O
pattern	O
analysis	O
,	O
whose	O
best	O
known	O
member	O
is	O
the	O
support-vector	B-Algorithm
machine	I-Algorithm
(	O
SVM	B-Algorithm
)	O
.	O
</s>
<s>
Kernel	B-Algorithm
methods	I-Algorithm
are	O
types	O
of	O
algorithms	O
that	O
are	O
used	O
for	O
pattern	O
analysis	O
.	O
</s>
<s>
These	O
methods	O
involve	O
using	O
linear	O
classifiers	B-General_Concept
to	O
solve	O
nonlinear	O
problems	O
.	O
</s>
<s>
The	O
general	O
task	O
of	O
pattern	O
analysis	O
is	O
to	O
find	O
and	O
study	O
general	O
types	O
of	O
relations	O
(	O
for	O
example	O
clusters	B-Algorithm
,	O
rankings	O
,	O
principal	B-Application
components	I-Application
,	O
correlations	O
,	O
classifications	B-General_Concept
)	O
in	O
datasets	O
.	O
</s>
<s>
For	O
many	O
algorithms	O
that	O
solve	O
these	O
tasks	O
,	O
the	O
data	O
in	O
raw	O
representation	O
have	O
to	O
be	O
explicitly	O
transformed	O
into	O
feature	B-Algorithm
vector	I-Algorithm
representations	O
via	O
a	O
user-specified	O
feature	O
map	O
:	O
in	O
contrast	O
,	O
kernel	B-Algorithm
methods	I-Algorithm
require	O
only	O
a	O
user-specified	O
kernel	O
,	O
i.e.	O
,	O
a	O
similarity	O
function	O
over	O
all	O
pairs	O
of	O
data	O
points	O
computed	O
using	O
inner	O
products	O
.	O
</s>
<s>
The	O
feature	O
map	O
in	O
kernel	B-Algorithm
machines	I-Algorithm
is	O
infinite	O
dimensional	O
but	O
only	O
requires	O
a	O
finite	O
dimensional	O
matrix	O
from	O
user-input	O
according	O
to	O
the	O
Representer	O
theorem	O
.	O
</s>
<s>
Kernel	B-Algorithm
machines	I-Algorithm
are	O
slow	O
to	O
compute	O
for	O
datasets	O
larger	O
than	O
a	O
couple	O
of	O
thousand	O
examples	O
without	O
parallel	O
processing	O
.	O
</s>
<s>
Kernel	B-Algorithm
methods	I-Algorithm
owe	O
their	O
name	O
to	O
the	O
use	O
of	O
kernel	O
functions	O
,	O
which	O
enable	O
them	O
to	O
operate	O
in	O
a	O
high-dimensional	O
,	O
implicit	O
feature	O
space	O
without	O
ever	O
computing	O
the	O
coordinates	O
of	O
the	O
data	O
in	O
that	O
space	O
,	O
but	O
rather	O
by	O
simply	O
computing	O
the	O
inner	O
products	O
between	O
the	O
images	O
of	O
all	O
pairs	O
of	O
data	O
in	O
the	O
feature	O
space	O
.	O
</s>
<s>
Kernel	O
functions	O
have	O
been	O
introduced	O
for	O
sequence	O
data	O
,	O
graphs	B-Algorithm
,	O
text	O
,	O
images	O
,	O
as	O
well	O
as	O
vectors	O
.	O
</s>
<s>
Algorithms	O
capable	O
of	O
operating	O
with	O
kernels	O
include	O
the	O
kernel	B-General_Concept
perceptron	I-General_Concept
,	O
support-vector	B-Algorithm
machines	I-Algorithm
(	O
SVM	B-Algorithm
)	O
,	O
Gaussian	B-General_Concept
processes	I-General_Concept
,	O
principal	B-Application
components	I-Application
analysis	I-Application
(	O
PCA	O
)	O
,	O
canonical	O
correlation	O
analysis	O
,	O
ridge	O
regression	O
,	O
spectral	B-Algorithm
clustering	I-Algorithm
,	O
linear	O
adaptive	O
filters	O
and	O
many	O
others	O
.	O
</s>
<s>
Typically	O
,	O
their	O
statistical	O
properties	O
are	O
analyzed	O
using	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
(	O
for	O
example	O
,	O
using	O
Rademacher	B-General_Concept
complexity	I-General_Concept
)	O
.	O
</s>
<s>
Kernel	B-Algorithm
methods	I-Algorithm
can	O
be	O
thought	O
of	O
as	O
instance-based	B-General_Concept
learners	I-General_Concept
:	O
rather	O
than	O
learning	O
some	O
fixed	O
set	O
of	O
parameters	O
corresponding	O
to	O
the	O
features	O
of	O
their	O
inputs	O
,	O
they	O
instead	O
"	O
remember	O
"	O
the	O
-th	O
training	O
example	O
and	O
learn	O
for	O
it	O
a	O
corresponding	O
weight	O
.	O
</s>
<s>
is	O
the	O
kernelized	O
binary	B-General_Concept
classifier	I-General_Concept
's	O
predicted	O
label	O
for	O
the	O
unlabeled	O
input	O
whose	O
hidden	O
true	O
label	O
is	O
of	O
interest	O
;	O
</s>
<s>
the	O
sum	O
ranges	O
over	O
the	O
labeled	O
examples	O
in	O
the	O
classifier	B-General_Concept
's	O
training	O
set	O
,	O
with	O
;	O
</s>
<s>
Kernel	O
classifiers	B-General_Concept
were	O
described	O
as	O
early	O
as	O
the	O
1960s	O
,	O
with	O
the	O
invention	O
of	O
the	O
kernel	B-General_Concept
perceptron	I-General_Concept
.	O
</s>
<s>
They	O
rose	O
to	O
great	O
prominence	O
with	O
the	O
popularity	O
of	O
the	O
support-vector	B-Algorithm
machine	I-Algorithm
(	O
SVM	B-Algorithm
)	O
in	O
the	O
1990s	O
,	O
when	O
the	O
SVM	B-Algorithm
was	O
found	O
to	O
be	O
competitive	O
with	O
neural	B-Architecture
networks	I-Architecture
on	O
tasks	O
such	O
as	O
handwriting	B-Application
recognition	I-Application
.	O
</s>
<s>
The	O
kernel	O
trick	O
avoids	O
the	O
explicit	O
mapping	O
that	O
is	O
needed	O
to	O
get	O
linear	O
learning	O
algorithms	O
to	O
learn	O
a	O
nonlinear	O
function	O
or	O
decision	B-General_Concept
boundary	I-General_Concept
.	O
</s>
<s>
Mercer	O
's	O
theorem	O
is	O
similar	O
to	O
a	O
generalization	O
of	O
the	O
result	O
from	O
linear	O
algebra	O
that	O
associates	O
an	O
inner	O
product	O
to	O
any	O
positive-definite	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Furthermore	O
,	O
there	O
is	O
often	O
no	O
need	O
to	O
compute	O
directly	O
during	O
computation	O
,	O
as	O
is	O
the	O
case	O
with	O
support-vector	B-Algorithm
machines	I-Algorithm
.	O
</s>
<s>
Theoretically	O
,	O
a	O
Gram	B-Algorithm
matrix	I-Algorithm
with	O
respect	O
to	O
(	O
sometimes	O
also	O
called	O
a	O
"	O
kernel	O
matrix	O
"	O
)	O
,	O
where	O
,	O
must	O
be	O
positive	O
semi-definite	O
(	O
PSD	O
)	O
.	O
</s>
<s>
If	O
the	O
kernel	O
function	O
is	O
also	O
a	O
covariance	O
function	O
as	O
used	O
in	O
Gaussian	B-General_Concept
processes	I-General_Concept
,	O
then	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
can	O
also	O
be	O
called	O
a	O
covariance	O
matrix	O
.	O
</s>
<s>
Application	O
areas	O
of	O
kernel	B-Algorithm
methods	I-Algorithm
are	O
diverse	O
and	O
include	O
geostatistics	O
,	O
kriging	O
,	O
inverse	O
distance	O
weighting	O
,	O
3D	B-Algorithm
reconstruction	I-Algorithm
,	O
bioinformatics	O
,	O
chemoinformatics	O
,	O
information	B-General_Concept
extraction	I-General_Concept
and	O
handwriting	B-Application
recognition	I-Application
.	O
</s>
