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SIR extracts the effective dimension reduction (
e.d.r
) subspace via using the
known class information as the responses in the regression formula. A noted semi-
supervised dimension reduction method was proposed in [13] which exploited
the pairwise constraint as the semi-supervised information and formulated as
an object function for optimization. Using pairwise constraint for dimension
reduction could be found in[10].
In this paper, the notation of data is defined as:
A
=[
x
1
;
;
x
n
]
n×p
be
···
∈
R
n
be the corresponding
response, the labels. In semi-supervised problems, large portion of
Y
is unknown
but fixed.
the data matrix of input attributes and
y
=[
y
1
;
...
;
y
n
]
∈
R
2 Supervised Kernel SIR for Dimension Reduction
Sliced inverse regression (SIR) [8] shows that the
e.d.r.
subspace can be estimated
from the leading directions, the most informative directions in the input pattern
space, in the central inverse regression function with the largest variation. SIR
finds the dimension reduction directions by solving the following generalized
eigenvalue problem:
ʣ
E
(
A|Y
J
)
ʲ
=
ʻʣ
A
ʲ,
(1)
where
ʣ
A
is the covariance matrix of
A
,
Y
J
denotes the membership in
J
slices,
and
ʣ
E
(
A|Y
J
)
denotes the between-slice covariance matrix based on sliced means
given by
J
ʣ
E
(
A|Y
J
)
=
1
n
x
)
.
n
j
(
x
j
x
)(
x
j
−
−
(2)
j
=1
i∈S
j
x
i
is the mean value of
the
j
th slice,
S
j
is the index set for
j
th slice,
n
j
is the size of
j
th slice. Note that
the slices are sliced from
A
according to responses
Y
.
In supervised problems,
x
j
is simply the class mean of input attributes for
the
j
th class in which the slices are replaced by the classes. An equivalent way
to modeling SIR by the following optimization problem:
n
i
=1
x
i
is the grand mean,
x
j
=
Here
x
=
1
1
n
j
ʲ
ʣ
E
(
A|Y
J
)
ʲ
subject to
ʲ
ʣ
A
ʲ
=1
max
β∈
R
p
(3)
The solution, denoted by
ʲ
1
, givens the first
e.d.r.
direction such that slice means
projected along
ʲ
1
are most spreading out, where
ʲ
1
is normalized with respect to
the sample covariance matrix
ʣ
A
. Repeatedly solving this optimization problem
with the orthogonality constraints
ʲ
k
ʣ
A
ʲ
l
=
ʴ
k,l
,
where
ʴ
k,l
is the Kronecker
delta, and the sequence of solution
ʲ
1
, ...ʲ
d
forms the
e.d.r.
basis. Some insightful
discussion on the SIR methodology and applications can be found in [12, 8].
Since the classical SIR is designed to find a linear transformation from input
space to a low dimensional subspace which retains as much information as pos-
sible for the output variable
y
, it may perform poorly in the non-linear tasks.
To solve the linearity problem, the kernel sliced inverse regression (KSIR)[11]
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