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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 ( r− 1) ], where A ( 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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