<s>
In	O
the	O
field	O
of	O
machine	O
learning	O
,	O
the	O
goal	O
of	O
statistical	B-General_Concept
classification	I-General_Concept
is	O
to	O
use	O
an	O
object	O
's	O
characteristics	O
to	O
identify	O
which	O
class	O
(	O
or	O
group	O
)	O
it	O
belongs	O
to	O
.	O
</s>
<s>
A	O
linear	B-General_Concept
classifier	I-General_Concept
achieves	O
this	O
by	O
making	O
a	O
classification	O
decision	O
based	O
on	O
the	O
value	O
of	O
a	O
linear	O
combination	O
of	O
the	O
characteristics	O
.	O
</s>
<s>
An	O
object	O
's	O
characteristics	O
are	O
also	O
known	O
as	O
feature	B-Algorithm
values	I-Algorithm
and	O
are	O
typically	O
presented	O
to	O
the	O
machine	O
in	O
a	O
vector	O
called	O
a	O
feature	B-Algorithm
vector	I-Algorithm
.	O
</s>
<s>
Such	O
classifiers	B-General_Concept
work	O
well	O
for	O
practical	O
problems	O
such	O
as	O
document	B-Algorithm
classification	I-Algorithm
,	O
and	O
more	O
generally	O
for	O
problems	O
with	O
many	O
variables	O
(	O
features	B-Algorithm
)	O
,	O
reaching	O
accuracy	O
levels	O
comparable	O
to	O
non-linear	O
classifiers	B-General_Concept
while	O
taking	O
less	O
time	O
to	O
train	O
and	O
use	O
.	O
</s>
<s>
(	O
In	O
other	O
words	O
,	O
is	O
a	O
one-form	B-Algorithm
or	O
linear	B-Algorithm
functional	I-Algorithm
mapping	O
onto	O
R	O
.	O
)	O
The	O
weight	O
vector	O
is	O
learned	O
from	O
a	O
set	O
of	O
labeled	O
training	O
samples	O
.	O
</s>
<s>
For	O
a	O
two-class	O
classification	O
problem	O
,	O
one	O
can	O
visualize	O
the	O
operation	O
of	O
a	O
linear	B-General_Concept
classifier	I-General_Concept
as	O
splitting	O
a	O
high-dimensional	O
input	O
space	O
with	O
a	O
hyperplane	O
:	O
all	O
points	O
on	O
one	O
side	O
of	O
the	O
hyperplane	O
are	O
classified	O
as	O
"	O
yes	O
"	O
,	O
while	O
the	O
others	O
are	O
classified	O
as	O
"	O
no	O
"	O
.	O
</s>
<s>
A	O
linear	B-General_Concept
classifier	I-General_Concept
is	O
often	O
used	O
in	O
situations	O
where	O
the	O
speed	O
of	O
classification	O
is	O
an	O
issue	O
,	O
since	O
it	O
is	O
often	O
the	O
fastest	O
classifier	B-General_Concept
,	O
especially	O
when	O
is	O
sparse	O
.	O
</s>
<s>
Also	O
,	O
linear	B-General_Concept
classifiers	I-General_Concept
often	O
work	O
very	O
well	O
when	O
the	O
number	O
of	O
dimensions	O
in	O
is	O
large	O
,	O
as	O
in	O
document	B-Algorithm
classification	I-Algorithm
,	O
where	O
each	O
element	O
in	O
is	O
typically	O
the	O
number	O
of	O
occurrences	O
of	O
a	O
word	O
in	O
a	O
document	O
(	O
see	O
document-term	B-General_Concept
matrix	I-General_Concept
)	O
.	O
</s>
<s>
In	O
such	O
cases	O
,	O
the	O
classifier	B-General_Concept
should	O
be	O
well-regularized	O
.	O
</s>
<s>
There	O
are	O
two	O
broad	O
classes	O
of	O
methods	O
for	O
determining	O
the	O
parameters	O
of	O
a	O
linear	B-General_Concept
classifier	I-General_Concept
.	O
</s>
<s>
Naive	B-General_Concept
Bayes	I-General_Concept
classifier	I-General_Concept
with	O
multinomial	O
or	O
multivariate	O
Bernoulli	O
event	O
models	O
.	O
</s>
<s>
Examples	O
of	O
discriminative	O
training	O
of	O
linear	B-General_Concept
classifiers	I-General_Concept
include	O
:	O
</s>
<s>
Logistic	O
regression	O
—	O
maximum	O
likelihood	O
estimation	O
of	O
assuming	O
that	O
the	O
observed	O
training	O
set	O
was	O
generated	O
by	O
a	O
binomial	O
model	O
that	O
depends	O
on	O
the	O
output	O
of	O
the	O
classifier	B-General_Concept
.	O
</s>
<s>
Fisher	B-General_Concept
's	I-General_Concept
Linear	I-General_Concept
Discriminant	I-General_Concept
Analysis	O
—	O
an	O
algorithm	O
(	O
different	O
than	O
"	O
LDA	O
"	O
)	O
that	O
maximizes	O
the	O
ratio	O
of	O
between-class	O
scatter	O
to	O
within-class	O
scatter	O
,	O
without	O
any	O
other	O
assumptions	O
.	O
</s>
<s>
It	O
is	O
in	O
essence	O
a	O
method	O
of	O
dimensionality	B-Algorithm
reduction	I-Algorithm
for	O
binary	O
classification	O
.	O
</s>
<s>
Support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
—	O
an	O
algorithm	O
that	O
maximizes	O
the	O
margin	B-Algorithm
between	O
the	O
decision	O
hyperplane	O
and	O
the	O
examples	O
in	O
the	O
training	O
set	O
.	O
</s>
<s>
However	O
,	O
its	O
name	O
makes	O
sense	O
when	O
we	O
compare	O
LDA	O
to	O
the	O
other	O
main	O
linear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
algorithm	O
:	O
principal	B-Application
components	I-Application
analysis	I-Application
(	O
PCA	O
)	O
.	O
</s>
<s>
LDA	O
is	O
a	O
supervised	B-General_Concept
learning	I-General_Concept
algorithm	O
that	O
utilizes	O
the	O
labels	O
of	O
the	O
data	O
,	O
while	O
PCA	O
is	O
an	O
unsupervised	B-General_Concept
learning	I-General_Concept
algorithm	O
that	O
ignores	O
the	O
labels	O
.	O
</s>
<s>
All	O
of	O
the	O
linear	B-General_Concept
classifier	I-General_Concept
algorithms	O
listed	O
above	O
can	O
be	O
converted	O
into	O
non-linear	O
algorithms	O
operating	O
on	O
a	O
different	O
input	O
space	O
,	O
using	O
the	O
kernel	O
trick	O
.	O
</s>
<s>
Discriminative	O
training	O
of	O
linear	B-General_Concept
classifiers	I-General_Concept
usually	O
proceeds	O
in	O
a	O
supervised	B-General_Concept
way	O
,	O
by	O
means	O
of	O
an	O
optimization	O
algorithm	O
that	O
is	O
given	O
a	O
training	O
set	O
with	O
desired	O
outputs	O
and	O
a	O
loss	O
function	O
that	O
measures	O
the	O
discrepancy	O
between	O
the	O
classifier	B-General_Concept
's	O
outputs	O
and	O
the	O
desired	O
outputs	O
.	O
</s>
<s>
is	O
a	O
vector	O
of	O
classifier	B-General_Concept
parameters	O
,	O
</s>
<s>
is	O
a	O
loss	O
function	O
that	O
measures	O
the	O
discrepancy	O
between	O
the	O
classifier	B-General_Concept
's	O
prediction	O
and	O
the	O
true	O
output	O
for	O
the	O
'	O
th	O
training	O
example	O
,	O
</s>
<s>
Popular	O
loss	O
functions	O
include	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
(	O
for	O
linear	O
SVMs	B-Algorithm
)	O
and	O
the	O
log	O
loss	O
(	O
for	O
linear	O
logistic	O
regression	O
)	O
.	O
</s>
<s>
Many	O
algorithms	O
exist	O
for	O
solving	O
such	O
problems	O
;	O
popular	O
ones	O
for	O
linear	B-General_Concept
classification	I-General_Concept
include	O
(	O
stochastic	B-Algorithm
)	O
gradient	B-Algorithm
descent	I-Algorithm
,	O
L-BFGS	B-Algorithm
,	O
coordinate	B-Algorithm
descent	I-Algorithm
and	O
Newton	O
methods	O
.	O
</s>
