2.2
The
Maximal
Separation
Model
We
seek
v, β
that
defines
the
hyperplane
H
v,β
:=
{
x

v
T
x
=
β
}
for
which:
•
v
T
a
i
> β
for
all
i
= 1
, . . . , k
•
v
T
b
i
< β
for
all
i
= 1
, . . . , m
We
would
like
the
hyperplane
H
v,β
not
only
to
separate
the
points
with
m
different
properties,
but
to
be
as
far
away
from
the
points
a
1
, . . . , a
k
, b
1
, . . . , b
as
possible.
It
is
easy
to
derive
via
elementary
analysis
that
the
distance
from
the
hyperplane
H
v,β
to
any
point
a
i
is
equal
to
v
T
a
i
−
β
.
v
Similarly,
the
distance
from
the
hyperplane
H
v,β
to
any
point
b
i
is
equal
to
T
b
i
β
−
v
.
v
If
we
normalize
the
vector
v
so
that
v
= 1
,
then
the
minimum
distance
from
the
hyperplane
H
v,β
to
any
of
the
points
a
1
, . . . , a
k
, b
1
, . . . , b
m
is
then:
T
T
b
1
m
min
v a
1
−
β, . . . , v
T
a
k
−
β, β
−
v
, . . . , β
−
v
T
b
.
We
therefore
would
like
v
and
β
to
satisfy:
•
v
= 1,
and
•
min
v
T
a
1
−
β, . . . , v
T
a
k
−
β, β
−
v
T
b
1
, . . . , β
−
v
T
b
m
is
maximized.
4