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T
n
( i b can be obtained
)
For any x , we have
p
(
x
)
=
c
y
(
x
)
. Each normal vector
n
from the derivatives of
p . Consider the derivative of
p n
( x
)
as follows,
p
(
x
)
n
n
n
T
(
i
)
(
i
)
T
(
j
)
p
(
x
)
=
=
x
b
=
b
x
b
.
n
x
x
i
i
j
i
i
T
(
l
)
(
j
)
(
l
)
For a point
x
S
,
b
x
=
0
for l i . It can be noted that there is only
l
j
i
T
one non-zero term in
p n
( x
)
(
l
)
(
i
)
(
j
)
(
l
)
for l = i . Then,
, i.e.
p
(
x
)
=
b
b
x
0
n
j
the normal vector of
S is yielded as,
(
l
)
p
(
x
)
b
(
i
)
=
n
.
(A2)
p
(
x
(
l
)
)
n
In order to get a set of points lying on each hyperplane respectively, so as to de-
termine the corresponding normal vectors
( i b , we can choose a point in the given
X close to one of the hyperplanes as follows,
)
px
()
n
i
=
arg
min
, where
x
X
.
(A3)
px
()
∇≠
px
()0
n
n
( i
)
After given the normal vectors
{
b
}
, we can classify the whole point set X into n
hyperplanes in
R as follows,
(
)
T
(
j
)
label
=
arg
min
x
b
,
j
=
1
...
n
.
(A4)
j
This algorithm is called GPCA-PDA Alg. in [12,13].
Polynomial Segmentation Algorithm
Consider a special case of piecewise constant data. Given N data points x œ R , we
hope to segment them into an unknown number of groups n . This implies that
there exist n unknown cluster centers
μ
≠≠
...
μ
, so that,
1
n
(
x
=
μ
)
∪∪
...
(
x
=
μ
)
,
1
n
which can be described in a polynomial form as follows,
n
n
.
k
px x
( )
=
(
μ
)
=
cx
=
0
(A5)
n
i
k
i
=
1
k
=
0
 
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