<s>
In	O
machine	O
learning	O
,	O
the	O
polynomial	B-Algorithm
kernel	I-Algorithm
is	O
a	O
kernel	O
function	O
commonly	O
used	O
with	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
(	O
SVMs	B-Algorithm
)	O
and	O
other	O
kernelized	O
models	O
,	O
that	O
represents	O
the	O
similarity	O
of	O
vectors	O
(	O
training	O
samples	O
)	O
in	O
a	O
feature	O
space	O
over	O
polynomials	O
of	O
the	O
original	O
variables	O
,	O
allowing	O
learning	O
of	O
non-linear	O
models	O
.	O
</s>
<s>
Intuitively	O
,	O
the	O
polynomial	B-Algorithm
kernel	I-Algorithm
looks	O
not	O
only	O
at	O
the	O
given	O
features	O
of	O
input	O
samples	O
to	O
determine	O
their	O
similarity	O
,	O
but	O
also	O
combinations	O
of	O
these	O
.	O
</s>
<s>
The	O
(	O
implicit	O
)	O
feature	O
space	O
of	O
a	O
polynomial	B-Algorithm
kernel	I-Algorithm
is	O
equivalent	O
to	O
that	O
of	O
polynomial	O
regression	O
,	O
but	O
without	O
the	O
combinatorial	O
blowup	O
in	O
the	O
number	O
of	O
parameters	O
to	O
be	O
learned	O
.	O
</s>
<s>
Although	O
the	O
RBF	B-Algorithm
kernel	I-Algorithm
is	O
more	O
popular	O
in	O
SVM	B-Algorithm
classification	O
than	O
the	O
polynomial	B-Algorithm
kernel	I-Algorithm
,	O
the	O
latter	O
is	O
quite	O
popular	O
in	O
natural	B-Language
language	I-Language
processing	I-Language
(	O
NLP	B-Language
)	O
.	O
</s>
<s>
The	O
most	O
common	O
degree	O
is	O
(	O
quadratic	O
)	O
,	O
since	O
larger	O
degrees	O
tend	O
to	O
overfit	B-Error_Name
on	O
NLP	B-Language
problems	O
.	O
</s>
<s>
Various	O
ways	O
of	O
computing	O
the	O
polynomial	B-Algorithm
kernel	I-Algorithm
(	O
both	O
exact	O
and	O
approximate	O
)	O
have	O
been	O
devised	O
as	O
alternatives	O
to	O
the	O
usual	O
non-linear	O
SVM	B-Algorithm
training	O
algorithms	O
,	O
including	O
:	O
</s>
<s>
full	O
expansion	O
of	O
the	O
kernel	O
prior	O
to	O
training/testing	O
with	O
a	O
linear	O
SVM	B-Algorithm
,	O
i.e.	O
</s>
<s>
basket	B-Algorithm
mining	I-Algorithm
(	O
using	O
a	O
variant	O
of	O
the	O
apriori	B-Algorithm
algorithm	I-Algorithm
)	O
for	O
the	O
most	O
commonly	O
occurring	O
feature	O
conjunctions	O
in	O
a	O
training	O
set	O
to	O
produce	O
an	O
approximate	O
expansion	O
;	O
</s>
<s>
One	O
problem	O
with	O
the	O
polynomial	B-Algorithm
kernel	I-Algorithm
is	O
that	O
it	O
may	O
suffer	O
from	O
numerical	B-Algorithm
instability	I-Algorithm
:	O
when	O
,	O
tends	O
to	O
zero	O
with	O
increasing	O
,	O
whereas	O
when	O
,	O
tends	O
to	O
infinity	O
.	O
</s>
