Information Technology Reference
In-Depth Information
∑
i
i
c
ii
y
row
min
ii
.
i
=
i
y
row
y
source
row
i
s
.
t
ii
+
=
1
∀
i
i
=
i
y
row
y
sink
row
ii
+
=
∀
1
i
i
∑
i
y
source
row
i
=
1
∑
i
y
sink
row
i
=
1
f
source
row
i
y
source
row
i
=
n
·
∀
i
i
(
f
row
f
row
f
source
row
i
f
sink
row
i
−
)+
−
=
1
∀
i
i
i
ii
f
row
ii
y
row
ii
i
)
≤
(
n
−
1
)
·
∀
(
i
,
f
row
ii
y
row
ii
i
)
≥
∀
(
i
,
y
row
y
source
row
i
y
sink
row
i
,
,
∈{
0
,
1
}.
ii
In the TSP model, the variables are the same as network flow model except including
variables
y
source
row
i
y
sink
row
i
,
, and the optimization problem is
∑
i
i
c
ii
y
row
min
ii
.
i
y
row
ii
s
.
t
=
1
∀
i
i
y
row
=
1
∀
i
i
i
y
row
ii
∈{
0
,
1
}.
The two optimization problems induced by the models are mixed integer linear
programming and can be solved by CPLEX [14].
After reordering the rows of data matrix, for rows
i
and
i
1 in the final
rearranged matrix, the median of each pairwise term of the objective function
φ
(
+
. In [19], top 10% of largest
median values are suggested to be boundaries between re-ordered rows.
nsNMF.
Pascual-Montano et al. [43] and Carmona-Saez et al. [8] proposed
a biclustering algorithm based on nonsmooth nonnegative matrix factorization
(nsNMF). The method nsNMF approximates the data matrix
A
as a product of two
submatrices,
W
and
H
.Rowsof
H
constitute basis samples, while columns of
W
are basis features. Coefficients in each pair of basis samples and features are used to
sort features and samples in the original matrix, respectively. The biclusters are the
submatrices of the sorted matrix.
Originally, the nonnegative matrix factorization is used to analyze facial im-
ages [35]. The nonnegative matrix factorization (NMF) is to decompose matrix
A
a
i
,
j
,
a
i
+
1
,
j
)
is computed by MEDIAN
j
φ
(
a
i
,
j
,
a
i
+
1
,
j
)
=(
a
ij
)
n
×
m
into two matrices, i.e.,
