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Assume that we further analyze the which denotes the k:th basis
B
k
(1
≤
k
r
). To keep the reconstruction of the original data, all the bases excluding
B
k
(number:r-1) in the root level should act as a part of
B
k
's sub-bases and the
others are new sub-bases (number: p). Hence, there are (r+p-1) sub-bases of
B
k
in all.
So, the sub-basis-matrix is
A
=[
A
n×p
,A
n×
(
r−
1)
], where
A
n×
(
r−
1)
consists of
all the columns excluding the k:th one in A. The sub-hidden-components-matrix
is
S
=[
S
p×m
S
(
r−
1)
×m
]
T
,where
S
(
r−
1)
×m
consists of all the rows excluding
the k:th one in S.
b) The basis
B
k
gives different contributions to each data, so the correspond-
ing sparseness of the data should be different in the process of further analyzing
B
k
. In other words, the data with larger hidden component on
B
k
should have
larger sparseness in the sublevel, which can be implemented through increasing
the sparseness regulatory parameter. Then, all the data related to
B
k
in the
upper level can be described more accurately.
Establishing objective function as follow:
≤
r
+
p
−
1
m
F
(
A,S
)=
1
2
+
λ
2
X
−
A S
S
kj
S
ij
(2)
j
=1
i
=1
where
λ
is the sparseness regulatory parameter. Then minimize the function
under the update rules:
1) Set t=0. Iterate the steps (2)-(4) until convergence.
2)
A
=
A
t
μ
(
A
t
S
X
)(
S
)
T
−
−
(3)
Any negative value in
A
is set to zero;
Rescale each column of
A
to unit norm, and then set
A
t
+1
=
A
.
3)
S
t
+1
=
S
t
.
((
A
t
+1
)
T
X
)
./
((
A
t
+1
)
T
(
A
t
+1
)
S
t
+
Λ
)
∗
(4)
where
.
and
./
denote element-wise multiplication and division (respectively),
and
Λ
is the sparseness regulatory parameters matrix namely
∗
1]
T
[
S
k
1
,S
k
2
,
Λ
=
λ
[1 1
···
···
,S
km
]
(5)
4) Increment t.
Consequently, we get the optimal sub-basis matrix
A
and sub-hidden-
components matrix
S
of the basis
B
k
.
c) Expanding and further analyzing the bases is illustrated in the following
figure:
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