<s>
In	O
machine	O
learning	O
,	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
(	O
SVMs	B-Algorithm
,	O
also	O
support	O
vector	O
networks	O
)	O
are	O
supervised	B-General_Concept
learning	I-General_Concept
models	O
with	O
associated	O
learning	O
algorithms	O
that	O
analyze	O
data	O
for	O
classification	B-General_Concept
and	O
regression	O
analysis	O
.	O
</s>
<s>
Developed	O
at	O
AT&T	O
Bell	O
Laboratories	O
by	O
Vladimir	O
Vapnik	O
with	O
colleagues	O
(	O
Boser	O
et	O
al.	O
,	O
1992	O
,	O
Guyon	O
et	O
al.	O
,	O
1993	O
,	O
Cortes	O
and	O
Vapnik	O
,	O
1995	O
,	O
Vapnik	O
et	O
al.	O
,	O
1997	O
)	O
SVMs	B-Algorithm
are	O
one	O
of	O
the	O
most	O
robust	O
prediction	O
methods	O
,	O
being	O
based	O
on	O
statistical	O
learning	O
frameworks	O
or	O
VC	O
theory	O
proposed	O
by	O
Vapnik	O
(	O
1982	O
,	O
1995	O
)	O
and	O
Chervonenkis	O
(	O
1974	O
)	O
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
training	O
examples	O
,	O
each	O
marked	O
as	O
belonging	O
to	O
one	O
of	O
two	O
categories	O
,	O
an	O
SVM	B-Algorithm
training	O
algorithm	O
builds	O
a	O
model	O
that	O
assigns	O
new	O
examples	O
to	O
one	O
category	O
or	O
the	O
other	O
,	O
making	O
it	O
a	O
non-probabilistic	O
binary	B-General_Concept
linear	B-General_Concept
classifier	I-General_Concept
(	O
although	O
methods	O
such	O
as	O
Platt	B-General_Concept
scaling	I-General_Concept
exist	O
to	O
use	O
SVM	B-Algorithm
in	O
a	O
probabilistic	B-General_Concept
classification	I-General_Concept
setting	O
)	O
.	O
</s>
<s>
SVM	B-Algorithm
maps	O
training	O
examples	O
to	O
points	O
in	O
space	O
so	O
as	O
to	O
maximise	O
the	O
width	O
of	O
the	O
gap	O
between	O
the	O
two	O
categories	O
.	O
</s>
<s>
In	O
addition	O
to	O
performing	O
linear	B-General_Concept
classification	I-General_Concept
,	O
SVMs	B-Algorithm
can	O
efficiently	O
perform	O
a	O
non-linear	O
classification	O
using	O
what	O
is	O
called	O
the	O
kernel	B-Algorithm
trick	O
,	O
implicitly	O
mapping	O
their	O
inputs	O
into	O
high-dimensional	O
feature	O
spaces	O
.	O
</s>
<s>
The	O
support	O
vector	O
clustering	B-Algorithm
algorithm	I-Algorithm
,	O
created	O
by	O
Hava	O
Siegelmann	O
and	O
Vladimir	O
Vapnik	O
,	O
applies	O
the	O
statistics	O
of	O
support	O
vectors	O
,	O
developed	O
in	O
the	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
algorithm	O
,	O
to	O
categorize	O
unlabeled	O
data	O
.	O
</s>
<s>
These	O
data	O
sets	O
require	O
unsupervised	B-General_Concept
learning	I-General_Concept
approaches	O
,	O
which	O
attempt	O
to	O
find	O
natural	O
clustering	B-Algorithm
of	I-Algorithm
the	I-Algorithm
data	I-Algorithm
to	O
groups	O
and	O
,	O
then	O
,	O
to	O
map	O
new	O
data	O
according	O
to	O
these	O
clusters	O
.	O
</s>
<s>
Classifying	B-General_Concept
data	I-General_Concept
is	O
a	O
common	O
task	O
in	O
machine	O
learning	O
.	O
</s>
<s>
Suppose	O
some	O
given	O
data	B-Application
points	I-Application
each	O
belong	O
to	O
one	O
of	O
two	O
classes	O
,	O
and	O
the	O
goal	O
is	O
to	O
decide	O
which	O
class	O
a	O
new	O
data	B-Application
point	I-Application
will	O
be	O
in	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
,	O
a	O
data	B-Application
point	I-Application
is	O
viewed	O
as	O
a	O
-dimensional	O
vector	O
(	O
a	O
list	O
of	O
numbers	O
)	O
,	O
and	O
we	O
want	O
to	O
know	O
whether	O
we	O
can	O
separate	O
such	O
points	O
with	O
a	O
-dimensional	O
hyperplane	O
.	O
</s>
<s>
This	O
is	O
called	O
a	O
linear	B-General_Concept
classifier	I-General_Concept
.	O
</s>
<s>
One	O
reasonable	O
choice	O
as	O
the	O
best	O
hyperplane	O
is	O
the	O
one	O
that	O
represents	O
the	O
largest	O
separation	O
,	O
or	O
margin	B-Algorithm
,	O
between	O
the	O
two	O
classes	O
.	O
</s>
<s>
So	O
we	O
choose	O
the	O
hyperplane	O
so	O
that	O
the	O
distance	O
from	O
it	O
to	O
the	O
nearest	O
data	B-Application
point	I-Application
on	O
each	O
side	O
is	O
maximized	O
.	O
</s>
<s>
If	O
such	O
a	O
hyperplane	O
exists	O
,	O
it	O
is	O
known	O
as	O
the	O
maximum-margin	O
hyperplane	O
and	O
the	O
linear	B-General_Concept
classifier	I-General_Concept
it	O
defines	O
is	O
known	O
as	O
a	O
maximum-margin	O
classifier	O
;	O
or	O
equivalently	O
,	O
the	O
perceptron	B-Algorithm
of	O
optimal	O
stability	O
.	O
</s>
<s>
More	O
formally	O
,	O
a	O
support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
constructs	O
a	O
hyperplane	O
or	O
set	O
of	O
hyperplanes	O
in	O
a	O
high	O
or	O
infinite-dimensional	O
space	O
,	O
which	O
can	O
be	O
used	O
for	O
classification	B-General_Concept
,	O
regression	O
,	O
or	O
other	O
tasks	O
like	O
outliers	O
detection	O
.	O
</s>
<s>
Intuitively	O
,	O
a	O
good	O
separation	O
is	O
achieved	O
by	O
the	O
hyperplane	O
that	O
has	O
the	O
largest	O
distance	O
to	O
the	O
nearest	O
training-data	O
point	O
of	O
any	O
class	O
(	O
so-called	O
functional	O
margin	B-Algorithm
)	O
,	O
since	O
in	O
general	O
the	O
larger	O
the	O
margin	B-Algorithm
,	O
the	O
lower	O
the	O
generalization	B-Algorithm
error	I-Algorithm
of	O
the	O
classifier	B-General_Concept
.	O
</s>
<s>
To	O
keep	O
the	O
computational	O
load	O
reasonable	O
,	O
the	O
mappings	O
used	O
by	O
SVM	B-Algorithm
schemes	O
are	O
designed	O
to	O
ensure	O
that	O
dot	O
products	O
of	O
pairs	O
of	O
input	O
data	O
vectors	O
may	O
be	O
computed	O
easily	O
in	O
terms	O
of	O
the	O
variables	O
in	O
the	O
original	O
space	O
,	O
by	O
defining	O
them	O
in	O
terms	O
of	O
a	O
kernel	B-Algorithm
function	O
selected	O
to	O
suit	O
the	O
problem	O
.	O
</s>
<s>
The	O
vectors	O
defining	O
the	O
hyperplanes	O
can	O
be	O
chosen	O
to	O
be	O
linear	O
combinations	O
with	O
parameters	O
of	O
images	O
of	O
feature	B-Algorithm
vectors	I-Algorithm
that	O
occur	O
in	O
the	O
data	O
base	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
the	O
sum	O
of	O
kernels	B-Algorithm
above	O
can	O
be	O
used	O
to	O
measure	O
the	O
relative	O
nearness	O
of	O
each	O
test	O
point	O
to	O
the	O
data	B-Application
points	I-Application
originating	O
in	O
one	O
or	O
the	O
other	O
of	O
the	O
sets	O
to	O
be	O
discriminated	O
.	O
</s>
<s>
SVMs	B-Algorithm
can	O
be	O
used	O
to	O
solve	O
various	O
real-world	O
problems	O
:	O
</s>
<s>
SVMs	B-Algorithm
are	O
helpful	O
in	O
text	B-Algorithm
and	I-Algorithm
hypertext	I-Algorithm
categorization	I-Algorithm
,	O
as	O
their	O
application	O
can	O
significantly	O
reduce	O
the	O
need	O
for	O
labeled	O
training	O
instances	O
in	O
both	O
the	O
standard	O
inductive	O
and	O
transductive	B-General_Concept
settings	O
.	O
</s>
<s>
Some	O
methods	O
for	O
shallow	O
semantic	O
parsing	O
are	O
based	O
on	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
.	O
</s>
<s>
Classification	B-General_Concept
of	O
images	O
can	O
also	O
be	O
performed	O
using	O
SVMs	B-Algorithm
.	O
</s>
<s>
Experimental	O
results	O
show	O
that	O
SVMs	B-Algorithm
achieve	O
significantly	O
higher	O
search	O
accuracy	O
than	O
traditional	O
query	O
refinement	O
schemes	O
after	O
just	O
three	O
to	O
four	O
rounds	O
of	O
relevance	O
feedback	O
.	O
</s>
<s>
This	O
is	O
also	O
true	O
for	O
image	B-Algorithm
segmentation	I-Algorithm
systems	O
,	O
including	O
those	O
using	O
a	O
modified	O
version	O
SVM	B-Algorithm
that	O
uses	O
the	O
privileged	O
approach	O
as	O
suggested	O
by	O
Vapnik	O
.	O
</s>
<s>
Classification	B-General_Concept
of	O
satellite	O
data	O
like	O
SAR	B-Application
data	O
using	O
supervised	O
SVM	B-Algorithm
.	O
</s>
<s>
Hand-written	O
characters	O
can	O
be	O
recognized	B-Application
using	O
SVM	B-Algorithm
.	O
</s>
<s>
The	O
SVM	B-Algorithm
algorithm	O
has	O
been	O
widely	O
applied	O
in	O
the	O
biological	O
and	O
other	O
sciences	O
.	O
</s>
<s>
Permutation	B-General_Concept
tests	I-General_Concept
based	O
on	O
SVM	B-Algorithm
weights	O
have	O
been	O
suggested	O
as	O
a	O
mechanism	O
for	O
interpretation	O
of	O
SVM	B-Algorithm
models	O
.	O
</s>
<s>
Support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
weights	O
have	O
also	O
been	O
used	O
to	O
interpret	O
SVM	B-Algorithm
models	O
in	O
the	O
past	O
.	O
</s>
<s>
Posthoc	O
interpretation	O
of	O
support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
models	O
in	O
order	O
to	O
identify	O
features	O
used	O
by	O
the	O
model	O
to	O
make	O
predictions	O
is	O
a	O
relatively	O
new	O
area	O
of	O
research	O
with	O
special	O
significance	O
in	O
the	O
biological	O
sciences	O
.	O
</s>
<s>
The	O
original	O
SVM	B-Algorithm
algorithm	O
was	O
invented	O
by	O
Vladimir	O
N	O
.	O
Vapnik	O
and	O
Alexey	O
Ya	O
.	O
</s>
<s>
In	O
1992	O
,	O
Bernhard	O
Boser	O
,	O
Isabelle	O
Guyon	O
and	O
Vladimir	O
Vapnik	O
suggested	O
a	O
way	O
to	O
create	O
nonlinear	O
classifiers	B-General_Concept
by	O
applying	O
the	O
kernel	B-Algorithm
trick	O
to	O
maximum-margin	O
hyperplanes	O
.	O
</s>
<s>
The	O
"	O
soft	O
margin	B-Algorithm
"	O
incarnation	O
,	O
as	O
is	O
commonly	O
used	O
in	O
software	O
packages	O
,	O
was	O
proposed	O
by	O
Corinna	O
Cortes	O
and	O
Vapnik	O
in	O
1993	O
and	O
published	O
in	O
1995	O
.	O
</s>
<s>
We	O
want	O
to	O
find	O
the	O
"	O
maximum-margin	O
hyperplane	O
"	O
that	O
divides	O
the	O
group	O
of	O
points	O
for	O
which	O
from	O
the	O
group	O
of	O
points	O
for	O
which	O
,	O
which	O
is	O
defined	O
so	O
that	O
the	O
distance	O
between	O
the	O
hyperplane	O
and	O
the	O
nearest	O
point	O
from	O
either	O
group	O
is	O
maximized	O
.	O
</s>
<s>
The	O
region	O
bounded	O
by	O
these	O
two	O
hyperplanes	O
is	O
called	O
the	O
"	O
margin	B-Algorithm
"	O
,	O
and	O
the	O
maximum-margin	O
hyperplane	O
is	O
the	O
hyperplane	O
that	O
lies	O
halfway	O
between	O
them	O
.	O
</s>
<s>
These	O
constraints	O
state	O
that	O
each	O
data	B-Application
point	I-Application
must	O
lie	O
on	O
the	O
correct	O
side	O
of	O
the	O
margin	B-Algorithm
.	O
</s>
<s>
The	O
and	O
that	O
solve	O
this	O
problem	O
determine	O
our	O
classifier	B-General_Concept
,	O
where	O
is	O
the	O
sign	O
function	O
.	O
</s>
<s>
An	O
important	O
consequence	O
of	O
this	O
geometric	O
description	O
is	O
that	O
the	O
max-margin	O
hyperplane	O
is	O
completely	O
determined	O
by	O
those	O
that	O
lie	O
nearest	O
to	O
it	O
.	O
</s>
<s>
This	O
function	O
is	O
zero	O
if	O
the	O
constraint	O
in	O
is	O
satisfied	O
,	O
in	O
other	O
words	O
,	O
if	O
lies	O
on	O
the	O
correct	O
side	O
of	O
the	O
margin	B-Algorithm
.	O
</s>
<s>
For	O
data	O
on	O
the	O
wrong	O
side	O
of	O
the	O
margin	B-Algorithm
,	O
the	O
function	O
's	O
value	O
is	O
proportional	O
to	O
the	O
distance	O
from	O
the	O
margin	B-Algorithm
.	O
</s>
<s>
where	O
the	O
parameter	O
determines	O
the	O
trade-off	O
between	O
increasing	O
the	O
margin	B-Algorithm
size	O
and	O
ensuring	O
that	O
the	O
lie	O
on	O
the	O
correct	O
side	O
of	O
the	O
margin	B-Algorithm
.	O
</s>
<s>
By	O
deconstructing	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
,	O
this	O
optimization	O
problem	O
can	O
be	O
massaged	O
into	O
the	O
following	O
:	O
</s>
<s>
Thus	O
,	O
for	O
large	O
values	O
of	O
,	O
it	O
will	O
behave	O
similar	O
to	O
the	O
hard-margin	O
SVM	B-Algorithm
,	O
if	O
the	O
input	O
data	O
are	O
linearly	O
classifiable	O
,	O
but	O
will	O
still	O
learn	O
if	O
a	O
classification	B-General_Concept
rule	O
is	O
viable	O
or	O
not	O
.	O
</s>
<s>
in	O
LIBSVM	B-Language
.	O
)	O
</s>
<s>
The	O
original	O
maximum-margin	O
hyperplane	O
algorithm	O
proposed	O
by	O
Vapnik	O
in	O
1963	O
constructed	O
a	O
linear	B-General_Concept
classifier	I-General_Concept
.	O
</s>
<s>
However	O
,	O
in	O
1992	O
,	O
Bernhard	O
Boser	O
,	O
Isabelle	O
Guyon	O
and	O
Vladimir	O
Vapnik	O
suggested	O
a	O
way	O
to	O
create	O
nonlinear	O
classifiers	B-General_Concept
by	O
applying	O
the	O
kernel	B-Algorithm
trick	O
(	O
originally	O
proposed	O
by	O
Aizerman	O
et	O
al	O
.	O
)	O
</s>
<s>
to	O
maximum-margin	O
hyperplanes	O
.	O
</s>
<s>
The	O
resulting	O
algorithm	O
is	O
formally	O
similar	O
,	O
except	O
that	O
every	O
dot	O
product	O
is	O
replaced	O
by	O
a	O
nonlinear	O
kernel	B-Algorithm
function	O
.	O
</s>
<s>
This	O
allows	O
the	O
algorithm	O
to	O
fit	O
the	O
maximum-margin	O
hyperplane	O
in	O
a	O
transformed	O
feature	O
space	O
.	O
</s>
<s>
The	O
transformation	O
may	O
be	O
nonlinear	O
and	O
the	O
transformed	O
space	O
high-dimensional	O
;	O
although	O
the	O
classifier	B-General_Concept
is	O
a	O
hyperplane	O
in	O
the	O
transformed	O
feature	O
space	O
,	O
it	O
may	O
be	O
nonlinear	O
in	O
the	O
original	O
input	O
space	O
.	O
</s>
<s>
It	O
is	O
noteworthy	O
that	O
working	O
in	O
a	O
higher-dimensional	O
feature	O
space	O
increases	O
the	O
generalization	B-Algorithm
error	I-Algorithm
of	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
,	O
although	O
given	O
enough	O
samples	O
the	O
algorithm	O
still	O
performs	O
well	O
.	O
</s>
<s>
Some	O
common	O
kernels	B-Algorithm
include	O
:	O
</s>
<s>
Polynomial	B-Algorithm
(	O
homogeneous	O
)	O
:	O
.	O
</s>
<s>
Particularly	O
,	O
when	O
,	O
this	O
becomes	O
the	O
linear	O
kernel	B-Algorithm
.	O
</s>
<s>
Polynomial	B-Algorithm
(	O
inhomogeneous	O
)	O
:	O
.	O
</s>
<s>
Gaussian	O
radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
:	O
for	O
.	O
</s>
<s>
Sigmoid	B-Algorithm
function	I-Algorithm
(	O
Hyperbolic	O
tangent	O
)	O
:	O
for	O
some	O
(	O
not	O
every	O
)	O
and	O
.	O
</s>
<s>
The	O
kernel	B-Algorithm
is	O
related	O
to	O
the	O
transform	O
by	O
the	O
equation	O
.	O
</s>
<s>
Dot	O
products	O
with	O
for	O
classification	B-General_Concept
can	O
again	O
be	O
computed	O
by	O
the	O
kernel	B-Algorithm
trick	O
,	O
i.e.	O
</s>
<s>
The	O
classical	O
approach	O
,	O
which	O
involves	O
reducing	O
to	O
a	O
quadratic	B-Algorithm
programming	I-Algorithm
problem	O
,	O
is	O
detailed	O
below	O
.	O
</s>
<s>
Then	O
,	O
more	O
recent	O
approaches	O
such	O
as	O
sub-gradient	B-Algorithm
descent	I-Algorithm
and	O
coordinate	B-Algorithm
descent	I-Algorithm
will	O
be	O
discussed	O
.	O
</s>
<s>
This	O
is	O
called	O
the	O
primal	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Since	O
the	O
dual	O
maximization	O
problem	O
is	O
a	O
quadratic	O
function	O
of	O
the	O
subject	O
to	O
linear	O
constraints	O
,	O
it	O
is	O
efficiently	O
solvable	O
by	O
quadratic	B-Algorithm
programming	I-Algorithm
algorithms	O
.	O
</s>
<s>
Moreover	O
,	O
exactly	O
when	O
lies	O
on	O
the	O
correct	O
side	O
of	O
the	O
margin	B-Algorithm
,	O
and	O
when	O
lies	O
on	O
the	O
margin	B-Algorithm
's	O
boundary	O
.	O
</s>
<s>
Suppose	O
now	O
that	O
we	O
would	O
like	O
to	O
learn	O
a	O
nonlinear	O
classification	B-General_Concept
rule	O
which	O
corresponds	O
to	O
a	O
linear	B-General_Concept
classification	I-General_Concept
rule	O
for	O
the	O
transformed	O
data	B-Application
points	I-Application
Moreover	O
,	O
we	O
are	O
given	O
a	O
kernel	B-Algorithm
function	O
which	O
satisfies	O
.	O
</s>
<s>
The	O
coefficients	O
can	O
be	O
solved	O
for	O
using	O
quadratic	B-Algorithm
programming	I-Algorithm
,	O
as	O
before	O
.	O
</s>
<s>
Recent	O
algorithms	O
for	O
finding	O
the	O
SVM	B-Algorithm
classifier	B-General_Concept
include	O
sub-gradient	B-Algorithm
descent	I-Algorithm
and	O
coordinate	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
Both	O
techniques	O
have	O
proven	O
to	O
offer	O
significant	O
advantages	O
over	O
the	O
traditional	O
approach	O
when	O
dealing	O
with	O
large	O
,	O
sparse	O
datasets	O
—	O
sub-gradient	O
methods	O
are	O
especially	O
efficient	O
when	O
there	O
are	O
many	O
training	O
examples	O
,	O
and	O
coordinate	B-Algorithm
descent	I-Algorithm
when	O
the	O
dimension	O
of	O
the	O
feature	O
space	O
is	O
high	O
.	O
</s>
<s>
As	O
such	O
,	O
traditional	O
gradient	B-Algorithm
descent	I-Algorithm
(	O
or	O
SGD	B-Algorithm
)	O
methods	O
can	O
be	O
adapted	O
,	O
where	O
instead	O
of	O
taking	O
a	O
step	O
in	O
the	O
direction	O
of	O
the	O
function	O
's	O
gradient	O
,	O
a	O
step	O
is	O
taken	O
in	O
the	O
direction	O
of	O
a	O
vector	O
selected	O
from	O
the	O
function	O
's	O
sub-gradient	O
.	O
</s>
<s>
This	O
approach	O
has	O
the	O
advantage	O
that	O
,	O
for	O
certain	O
implementations	O
,	O
the	O
number	O
of	O
iterations	O
does	O
not	O
scale	O
with	O
,	O
the	O
number	O
of	O
data	B-Application
points	I-Application
.	O
</s>
<s>
The	O
soft-margin	O
support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
described	O
above	O
is	O
an	O
example	O
of	O
an	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
(	O
ERM	O
)	O
algorithm	O
for	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
.	O
</s>
<s>
Seen	O
this	O
way	O
,	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
belong	O
to	O
a	O
natural	O
class	O
of	O
algorithms	O
for	O
statistical	O
inference	O
,	O
and	O
many	O
of	O
its	O
unique	O
features	O
are	O
due	O
to	O
the	O
behavior	O
of	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
.	O
</s>
<s>
This	O
perspective	O
can	O
provide	O
further	O
insight	O
into	O
how	O
and	O
why	O
SVMs	B-Algorithm
work	O
,	O
and	O
allow	O
us	O
to	O
better	O
analyze	O
their	O
statistical	O
properties	O
.	O
</s>
<s>
In	O
supervised	B-General_Concept
learning	I-General_Concept
,	O
one	O
is	O
given	O
a	O
set	O
of	O
training	O
examples	O
with	O
labels	O
,	O
and	O
wishes	O
to	O
predict	O
given	O
.	O
</s>
<s>
This	O
approach	O
is	O
called	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
,	O
or	O
ERM	O
.	O
</s>
<s>
If	O
is	O
a	O
normed	O
space	O
(	O
as	O
is	O
the	O
case	O
for	O
SVM	B-Algorithm
)	O
,	O
a	O
particularly	O
effective	O
technique	O
is	O
to	O
consider	O
only	O
those	O
hypotheses	O
for	O
which	O
.	O
</s>
<s>
Recall	O
that	O
the	O
(	O
soft-margin	O
)	O
SVM	B-Algorithm
classifier	B-General_Concept
is	O
chosen	O
to	O
minimize	O
the	O
following	O
expression	O
:	O
</s>
<s>
From	O
this	O
perspective	O
,	O
SVM	B-Algorithm
is	O
closely	O
related	O
to	O
other	O
fundamental	O
classification	B-General_Concept
algorithms	O
such	O
as	O
regularized	O
least-squares	O
and	O
logistic	O
regression	O
.	O
</s>
<s>
The	O
difference	O
between	O
the	O
three	O
lies	O
in	O
the	O
choice	O
of	O
loss	O
function	O
:	O
regularized	O
least-squares	O
amounts	O
to	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
with	O
the	O
square-loss	B-Algorithm
,	O
;	O
logistic	O
regression	O
employs	O
the	O
log-loss	B-Algorithm
,	O
</s>
<s>
The	O
difference	O
between	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
and	O
these	O
other	O
loss	O
functions	O
is	O
best	O
stated	O
in	O
terms	O
of	O
target	O
functions	O
-	O
the	O
function	O
that	O
minimizes	O
expected	O
risk	O
for	O
a	O
given	O
pair	O
of	O
random	O
variables	O
.	O
</s>
<s>
In	O
the	O
classification	B-General_Concept
setting	O
,	O
we	O
have	O
:	O
</s>
<s>
The	O
optimal	O
classifier	B-General_Concept
is	O
therefore	O
:	O
</s>
<s>
For	O
the	O
square-loss	B-Algorithm
,	O
the	O
target	O
function	O
is	O
the	O
conditional	O
expectation	O
function	O
,	O
;	O
For	O
the	O
logistic	O
loss	O
,	O
it	O
's	O
the	O
logit	O
function	O
,	O
.	O
</s>
<s>
While	O
both	O
of	O
these	O
target	O
functions	O
yield	O
the	O
correct	O
classifier	B-General_Concept
,	O
as	O
,	O
they	O
give	O
us	O
more	O
information	O
than	O
we	O
need	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
one	O
can	O
check	O
that	O
the	O
target	O
function	O
for	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
is	O
exactly	O
.	O
</s>
<s>
Thus	O
,	O
in	O
a	O
sufficiently	O
rich	O
hypothesis	O
space	O
—	O
or	O
equivalently	O
,	O
for	O
an	O
appropriately	O
chosen	O
kernel	B-Algorithm
—	O
the	O
SVM	B-Algorithm
classifier	B-General_Concept
will	O
converge	O
to	O
the	O
simplest	O
function	O
(	O
in	O
terms	O
of	O
)	O
that	O
correctly	O
classifies	O
the	O
data	O
.	O
</s>
<s>
This	O
extends	O
the	O
geometric	O
interpretation	O
of	O
SVM	B-Algorithm
—	O
for	O
linear	B-General_Concept
classification	I-General_Concept
,	O
the	O
empirical	O
risk	O
is	O
minimized	O
by	O
any	O
function	O
whose	O
margins	O
lie	O
between	O
the	O
support	O
vectors	O
,	O
and	O
the	O
simplest	O
of	O
these	O
is	O
the	O
max-margin	O
classifier	O
.	O
</s>
<s>
SVMs	B-Algorithm
belong	O
to	O
a	O
family	O
of	O
generalized	O
linear	B-General_Concept
classifiers	I-General_Concept
and	O
can	O
be	O
interpreted	O
as	O
an	O
extension	O
of	O
the	O
perceptron	B-Algorithm
.	O
</s>
<s>
A	O
special	O
property	O
is	O
that	O
they	O
simultaneously	O
minimize	O
the	O
empirical	O
classification	B-General_Concept
error	O
and	O
maximize	O
the	O
geometric	O
margin	B-Algorithm
;	O
hence	O
they	O
are	O
also	O
known	O
as	O
maximum	O
margin	B-General_Concept
classifiers	I-General_Concept
.	O
</s>
<s>
A	O
comparison	O
of	O
the	O
SVM	B-Algorithm
to	O
other	O
classifiers	B-General_Concept
has	O
been	O
made	O
by	O
Meyer	O
,	O
Leisch	O
and	O
Hornik	O
.	O
</s>
<s>
The	O
effectiveness	O
of	O
SVM	B-Algorithm
depends	O
on	O
the	O
selection	O
of	O
kernel	B-Algorithm
,	O
the	O
kernel	B-Algorithm
's	O
parameters	O
,	O
and	O
soft	O
margin	B-Algorithm
parameter	O
.	O
</s>
<s>
A	O
common	O
choice	O
is	O
a	O
Gaussian	O
kernel	B-Algorithm
,	O
which	O
has	O
a	O
single	O
parameter	O
.	O
</s>
<s>
Typically	O
,	O
each	O
combination	O
of	O
parameter	O
choices	O
is	O
checked	O
using	O
cross	B-Application
validation	I-Application
,	O
and	O
the	O
parameters	O
with	O
best	O
cross-validation	B-Application
accuracy	O
are	O
picked	O
.	O
</s>
<s>
Potential	O
drawbacks	O
of	O
the	O
SVM	B-Algorithm
include	O
the	O
following	O
aspects	O
:	O
</s>
<s>
The	O
SVM	B-Algorithm
is	O
only	O
directly	O
applicable	O
for	O
two-class	O
tasks	O
.	O
</s>
<s>
Therefore	O
,	O
algorithms	O
that	O
reduce	O
the	O
multi-class	O
task	O
to	O
several	O
binary	B-General_Concept
problems	O
have	O
to	O
be	O
applied	O
;	O
see	O
the	O
multi-class	O
SVM	B-Algorithm
section	O
.	O
</s>
<s>
SVC	O
is	O
a	O
similar	O
method	O
that	O
also	O
builds	O
on	O
kernel	B-Algorithm
functions	O
but	O
is	O
appropriate	O
for	O
unsupervised	B-General_Concept
learning	I-General_Concept
.	O
</s>
<s>
Multiclass	O
SVM	B-Algorithm
aims	O
to	O
assign	O
labels	O
to	O
instances	O
by	O
using	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
,	O
where	O
the	O
labels	O
are	O
drawn	O
from	O
a	O
finite	O
set	O
of	O
several	O
elements	O
.	O
</s>
<s>
The	O
dominant	O
approach	O
for	O
doing	O
so	O
is	O
to	O
reduce	O
the	O
single	O
multiclass	B-General_Concept
problem	I-General_Concept
into	O
multiple	O
binary	B-General_Concept
classification	I-General_Concept
problems	O
.	O
</s>
<s>
Building	O
binary	B-General_Concept
classifiers	I-General_Concept
that	O
distinguish	O
between	O
one	O
of	O
the	O
labels	O
and	O
the	O
rest	O
(	O
one-versus-all	O
)	O
or	O
between	O
every	O
pair	O
of	O
classes	O
(	O
one-versus-one	O
)	O
.	O
</s>
<s>
Classification	B-General_Concept
of	O
new	O
instances	O
for	O
the	O
one-versus-all	O
case	O
is	O
done	O
by	O
a	O
winner-takes-all	O
strategy	O
,	O
in	O
which	O
the	O
classifier	B-General_Concept
with	O
the	O
highest-output	O
function	O
assigns	O
the	O
class	O
(	O
it	O
is	O
important	O
that	O
the	O
output	O
functions	O
be	O
calibrated	O
to	O
produce	O
comparable	O
scores	O
)	O
.	O
</s>
<s>
For	O
the	O
one-versus-one	O
approach	O
,	O
classification	B-General_Concept
is	O
done	O
by	O
a	O
max-wins	O
voting	O
strategy	O
,	O
in	O
which	O
every	O
classifier	B-General_Concept
assigns	O
the	O
instance	O
to	O
one	O
of	O
the	O
two	O
classes	O
,	O
then	O
the	O
vote	O
for	O
the	O
assigned	O
class	O
is	O
increased	O
by	O
one	O
vote	O
,	O
and	O
finally	O
the	O
class	O
with	O
the	O
most	O
votes	O
determines	O
the	O
instance	O
classification	B-General_Concept
.	O
</s>
<s>
Crammer	O
and	O
Singer	O
proposed	O
a	O
multiclass	O
SVM	B-Algorithm
method	O
which	O
casts	O
the	O
multiclass	B-General_Concept
classification	I-General_Concept
problem	O
into	O
a	O
single	O
optimization	O
problem	O
,	O
rather	O
than	O
decomposing	O
it	O
into	O
multiple	O
binary	B-General_Concept
classification	I-General_Concept
problems	O
.	O
</s>
<s>
Transductive	B-Algorithm
support	I-Algorithm
vector	I-Algorithm
machines	I-Algorithm
extend	O
SVMs	B-Algorithm
in	O
that	O
they	O
could	O
also	O
treat	O
partially	O
labeled	O
data	O
in	O
semi-supervised	B-General_Concept
learning	I-General_Concept
by	O
following	O
the	O
principles	O
of	O
transduction	B-General_Concept
.	O
</s>
<s>
Formally	O
,	O
a	O
transductive	B-Algorithm
support	I-Algorithm
vector	I-Algorithm
machine	I-Algorithm
is	O
defined	O
by	O
the	O
following	O
primal	O
optimization	O
problem	O
:	O
</s>
<s>
Transductive	B-Algorithm
support	I-Algorithm
vector	I-Algorithm
machines	I-Algorithm
were	O
introduced	O
by	O
Vladimir	O
N	O
.	O
Vapnik	O
in	O
1998	O
.	O
</s>
<s>
SVMs	B-Algorithm
have	O
been	O
generalized	O
to	O
structured	B-Algorithm
SVMs	I-Algorithm
,	O
where	O
the	O
label	O
space	O
is	O
structured	O
and	O
of	O
possibly	O
infinite	O
size	O
.	O
</s>
<s>
A	O
version	O
of	O
SVM	B-Algorithm
for	O
regression	O
was	O
proposed	O
in	O
1996	O
by	O
Vladimir	O
N	O
.	O
Vapnik	O
,	O
Harris	O
Drucker	O
,	O
Christopher	O
J	O
.	O
C	O
.	O
Burges	O
,	O
Linda	O
Kaufman	O
and	O
Alexander	O
J	O
.	O
Smola	O
.	O
</s>
<s>
The	O
model	O
produced	O
by	O
support	O
vector	O
classification	B-General_Concept
(	O
as	O
described	O
above	O
)	O
depends	O
only	O
on	O
a	O
subset	O
of	O
the	O
training	O
data	O
,	O
because	O
the	O
cost	O
function	O
for	O
building	O
the	O
model	O
does	O
not	O
care	O
about	O
training	O
points	O
that	O
lie	O
beyond	O
the	O
margin	B-Algorithm
.	O
</s>
<s>
Another	O
SVM	B-Algorithm
version	O
known	O
as	O
least-squares	B-Algorithm
support	I-Algorithm
vector	I-Algorithm
machine	I-Algorithm
(	O
LS-SVM	O
)	O
has	O
been	O
proposed	O
by	O
Suykens	O
and	O
Vandewalle	O
.	O
</s>
<s>
In	O
2011	O
it	O
was	O
shown	O
by	O
Polson	O
and	O
Scott	O
that	O
the	O
SVM	B-Algorithm
admits	O
a	O
Bayesian	O
interpretation	O
through	O
the	O
technique	O
of	O
data	B-General_Concept
augmentation	I-General_Concept
.	O
</s>
<s>
In	O
this	O
approach	O
the	O
SVM	B-Algorithm
is	O
viewed	O
as	O
a	O
graphical	O
model	O
(	O
where	O
the	O
parameters	O
are	O
connected	O
via	O
probability	O
distributions	O
)	O
.	O
</s>
<s>
This	O
extended	O
view	O
allows	O
the	O
application	O
of	O
Bayesian	O
techniques	O
to	O
SVMs	B-Algorithm
,	O
such	O
as	O
flexible	O
feature	O
modeling	O
,	O
automatic	O
hyperparameter	B-General_Concept
tuning	O
,	O
and	O
predictive	O
uncertainty	O
quantification	O
.	O
</s>
<s>
Recently	O
,	O
a	O
scalable	O
version	O
of	O
the	O
Bayesian	O
SVM	B-Algorithm
was	O
developed	O
by	O
,	O
enabling	O
the	O
application	O
of	O
Bayesian	O
SVMs	B-Algorithm
to	O
big	B-Application
data	I-Application
.	O
</s>
<s>
Florian	O
Wenzel	O
developed	O
two	O
different	O
versions	O
,	O
a	O
variational	O
inference	O
(	O
VI	O
)	O
scheme	O
for	O
the	O
Bayesian	O
kernel	B-Algorithm
support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
(	O
SVM	B-Algorithm
)	O
and	O
a	O
stochastic	O
version	O
(	O
SVI	O
)	O
for	O
the	O
linear	O
Bayesian	O
SVM	B-Algorithm
.	O
</s>
<s>
The	O
parameters	O
of	O
the	O
maximum-margin	O
hyperplane	O
are	O
derived	O
by	O
solving	O
the	O
optimization	O
.	O
</s>
<s>
There	O
exist	O
several	O
specialized	O
algorithms	O
for	O
quickly	O
solving	O
the	O
quadratic	B-Algorithm
programming	I-Algorithm
(	O
QP	O
)	O
problem	O
that	O
arises	O
from	O
SVMs	B-Algorithm
,	O
mostly	O
relying	O
on	O
heuristics	O
for	O
breaking	O
the	O
problem	O
down	O
into	O
smaller	O
,	O
more	O
manageable	O
chunks	O
.	O
</s>
<s>
Another	O
approach	O
is	O
to	O
use	O
an	O
interior-point	B-Algorithm
method	I-Algorithm
that	O
uses	O
Newton-like	O
iterations	O
to	O
find	O
a	O
solution	O
of	O
the	O
Karush	O
–	O
Kuhn	O
–	O
Tucker	O
conditions	O
of	O
the	O
primal	O
and	O
dual	O
problems	O
.	O
</s>
<s>
To	O
avoid	O
solving	O
a	O
linear	O
system	O
involving	O
the	O
large	O
kernel	B-Algorithm
matrix	O
,	O
a	O
low-rank	O
approximation	O
to	O
the	O
matrix	O
is	O
often	O
used	O
in	O
the	O
kernel	B-Algorithm
trick	O
.	O
</s>
<s>
Another	O
common	O
method	O
is	O
Platt	O
's	O
sequential	B-Algorithm
minimal	I-Algorithm
optimization	I-Algorithm
(	O
SMO	O
)	O
algorithm	O
,	O
which	O
breaks	O
the	O
problem	O
down	O
into	O
2-dimensional	O
sub-problems	O
that	O
are	O
solved	O
analytically	O
,	O
eliminating	O
the	O
need	O
for	O
a	O
numerical	O
optimization	O
algorithm	O
and	O
matrix	O
storage	O
.	O
</s>
<s>
This	O
algorithm	O
is	O
conceptually	O
simple	O
,	O
easy	O
to	O
implement	O
,	O
generally	O
faster	O
,	O
and	O
has	O
better	O
scaling	O
properties	O
for	O
difficult	O
SVM	B-Algorithm
problems	O
.	O
</s>
<s>
The	O
special	O
case	O
of	O
linear	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
can	O
be	O
solved	O
more	O
efficiently	O
by	O
the	O
same	O
kind	O
of	O
algorithms	O
used	O
to	O
optimize	O
its	O
close	O
cousin	O
,	O
logistic	O
regression	O
;	O
this	O
class	O
of	O
algorithms	O
includes	O
sub-gradient	B-Algorithm
descent	I-Algorithm
(	O
e.g.	O
,	O
PEGASOS	O
)	O
and	O
coordinate	B-Algorithm
descent	I-Algorithm
(	O
e.g.	O
,	O
LIBLINEAR	B-Language
)	O
.	O
</s>
<s>
LIBLINEAR	B-Language
has	O
some	O
attractive	O
training-time	O
properties	O
.	O
</s>
<s>
Each	O
convergence	O
iteration	O
takes	O
time	O
linear	O
in	O
the	O
time	O
taken	O
to	O
read	O
the	O
train	O
data	O
,	O
and	O
the	O
iterations	O
also	O
have	O
a	O
Q-linear	B-Architecture
convergence	I-Architecture
property	O
,	O
making	O
the	O
algorithm	O
extremely	O
fast	O
.	O
</s>
<s>
The	O
general	O
kernel	B-Algorithm
SVMs	B-Algorithm
can	O
also	O
be	O
solved	O
more	O
efficiently	O
using	O
sub-gradient	B-Algorithm
descent	I-Algorithm
(	O
e.g.	O
</s>
<s>
P-packSVM	O
)	O
,	O
especially	O
when	O
parallelization	B-Operating_System
is	O
allowed	O
.	O
</s>
<s>
Kernel	B-Algorithm
SVMs	B-Algorithm
are	O
available	O
in	O
many	O
machine-learning	O
toolkits	O
,	O
including	O
LIBSVM	B-Language
,	O
MATLAB	B-Language
,	O
,	O
SVMlight	O
,	O
,	O
scikit-learn	B-Application
,	O
Shogun	B-Language
,	O
Weka	B-Language
,	O
,	O
,	O
OpenCV	B-Language
and	O
others	O
.	O
</s>
<s>
Preprocessing	O
of	O
data	O
(	O
standardization	O
)	O
is	O
highly	O
recommended	O
to	O
enhance	O
accuracy	O
of	O
classification	B-General_Concept
.	O
</s>
<s>
Subtraction	O
of	O
mean	O
and	O
division	O
by	O
variance	O
of	O
each	O
feature	O
is	O
usually	O
used	O
for	O
SVM	B-Algorithm
.	O
</s>
