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characteristics of words and the sparse part can indicate the specific characteristics of
words. In other words, we obtain two word-level distributions that can describe the
generality and specialty of words through low-rank and sparse matrix decomposition.
As mentioned above in Section 3.1,
W
= [
w
1
, w
2
, …, w
N
] is the collection of BoW
representations where
w
i
is the representation of
i-
th training image. Thus the decom-
position is defined as:
W
=
L
+
S
(4)
where
L
and
N
are the low-rank matrix and the sparse matrix. The problem of low-
rank and sparse matrix decomposition can be characterized by
mi
N
rank
(
L
)
+
ʻ
S
0
.
L,
(5)
s.t.
W
=
L
+
S
⇅
The
is zero-norm that counts the non-zero elements in the matrix and
0
ʻ
is the coefficient that balances the rank term and the sparsity term. Since the
problem is non-convex and hard to solve, it can be approximated by solving (6) ac-
cording to [13]:
>
0
mi
N
L
+
ʻ
S
*
1
L,
(6)
s.t.
W
=
L
+
S
⇅
The
is the nuclear norm defined as the sum of all singular values. The
*
problem can be solved by the augmented Lagrange multiplier method (ALM) pro-
posed by Lin et al [14].
3.3
Low-Rank and Sparse Matrix Decomposition Based pLSA Model
After the low-rank and sparse matrix decomposition, we obtain a low-rank matrix
L
which can characterize the correlated part of words and a sparse matrix
N
which
can characterize the specific part of words. Each column vector
l
i
of the matrix
L
=
[
l
1
,
l
2
, …,
l
N
] can be regarded as the representation of correlated characteristics of
the
i-
th training image, and each column vector
n
i
of the matrix
S
=
[
s
1
, s
2
, …,
s
N
]
implies the specific characteristics. Therefore, instead of
w
i
, we respectively apply
l
i
and
n
i
for the word-level representations of
i-
th image, and then use them to train two
pLSA models. Note that
l
i
and
n
i
are
L
1-
normalized after absolute operation.
The flow chart of our work is presented in Fig. 3. First we extract the 59-
dimensional block LBP histogram for each pathology images in the training set. Then
the codebook can be gained through
k
-means and the word-level representation
(namely
w
i
in matrix
W
) corresponding to each image is quantized. After the low-
rank and sparse matrix decomposition step, the matrices
L
and
S
take place of
W
to
be the word-level representations. EM algorithm is respectively used to compute the
optimal
P(z|d) and P(w|z) of these two representations, and the combination of P(z|d)
is the final topic-level representation of each image. In the test stage, the input ROI
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