![](https://csdnimg.cn/release/download_crawler_static/19157255/bg4.jpg)
IGðXjYÞ¼ENðXÞENðXjYÞð3Þ
And symmetrical uncertainty [26] is obtained by
normalizing IG(X|Y):
UðX; YÞ¼
IGðXjYÞ
ENðXÞþENðYÞ
ð4Þ
Based on the symmetrical uncertainty, given pairwise
groups S and S
0
, the inter-group feature correlation is
calculated as:
CðS; S
0
Þ¼
1
jSjjS
0
j
X
X2S;Y2S
0
UðX; YÞð5Þ
where
1
jSjjS
0
j
is a normalization factor to make
CðS; S
0
Þ2½1; 1. The intra-group feature correlation is
defined as:
CðSÞ¼CðS; SÞð6Þ
Based on (5) and (6), a clustering criterion is defined for
feature rearrangement. This criterion minimizes the inter-
group feature correlation and simultaneously maximizes
the intra-group feature correlation, i.e.,
min
X
L
i¼1;j [ i
CðS
i
; S
j
Þ
CðS
i
Þ
þ
CðS
i
; S
j
Þ
CðS
j
Þ
ð7Þ
(7) can be solved in an iterative way. Particularly, in the i-
th iteration, the i-th feature is allocated into a group. It is
noticeable that, as an entropy-based criterion, (7) only
measures nominal features directly. To describe continuous
features, it is necessary to discretize them into nominal
features beforehand.
After constructing L groups S¼fS
1
; S
2
; ...; S
L
g,we
map it onto the Grassmann manifold. Specifically, we align
the dimension of group S
i
to min(S
i
) using principle com-
ponent analysis (PCA) [27], and accordingly construct a
matrix M ¼½
e
S
1
;
e
S
2
; ...;
f
S
L
, where
e
S
i
is an aligned group.
Following the concept of the Grassmann manifold, M can
be deemed as a point on the Grassmann manifold. The
algorithm of projecting multiple features onto the Grass-
mann manifold is presented in Table 2.
3 Kernel matrix on the Grassmann manifold
3.1 Principle angles on the Grassmann manifold
As illustrated in Fig. 2, the Grassmann manifold G(L, Q)is
formed by the set of L-dimensional linear subspaces on R
Q
,
each corresponds to a matrix constructed based on Algo-
rithm 1. Let M; M
0
be a pair of same-sized matrices, there
are L principle angles between them, and the i-th principle
angle is calculated as:
Table 1 Important notations used in the paper
Notation Description
fX
1
; X
2
; ...; X
K
g Multiple features
fS
1
; S
2
; ...; S
L
g L groups rearranged multiple features
f
~
S
1
;
~
S
2
; ...;
~
S
L
g
L groups rearranged multiple features after PCA
M Matrix constructed by L groups rearranged
multiple features
x
i
Value of feature X
i
P(x
i
) The probability of x
i
existing in all training
samples
P(x
i
|y
i
) The probability of x
i
existing in all training
samples, conditioned on y
i
existing in all
training samples
EN(X) Entropy of feature X
EN(X|Y) Conditional entropy of feature Y, provided
feature X
IG(X|Y) Information gain
U(X,Y) Symmetrical uncertainty
C(S) Intra-group feature correlation
CðS; S
0
Þ Inter-group feature correlation
G(L, Q) Grassmann manifold by L-dimensional linear
subspace on R
Q
h
i
The i-th principle angle
b(M
i
) Orthonormal basis of matrix M
i
d
p
The projection distance
d
bc
The Binet–Cauchy distance
d
l2
The L
2
-Hausdorff distance
DA(Z) Discriminative ability of implicit feature Z
z
0
The value of implicit feature Z
W The N 9 N local structure matrix
R Diagonal matrix of W
LL= R - W
LS
Laplacian score: LSðZÞ¼~z
T
L~z=~z
T
R~z
CA Clustering accuracy
C One cluster
MI Mutual information
f
MI
Normalized mutual information
Table 2 Projecting multiple features onto the Grassmann manifold
(Algorithm 1)
L. Zhang et al.
123