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6. Nonoverlapping with tree-structured biclusters. For a data matrix
A
, nonoverlap-
ping means no entry can belong to more than one bicluster. Thus some entries
may not belong to any bicluster. Tree structure means in the proper rearrange-
ment matrix, the blocks of submatrices (biclusters) are not crossing each other.
7. Nonoverlapping nonexclusive biclusters. Nonoverlapping is same as above. Non-
exclusive means a sample or feature can belong to more than one biclusters, and
a sample can belong to two sets of important features in two biclusters, and vice
versa.
8. Nonoverlapping hierarchically structured biclusters. Nonoverlapping is same as
above. Hierarchically structured means a bicluster may belong to some other
“bigger” biclusters, i.e., in the set of biclusters
B
=
{
B
k
=(
S
k
,F
k
)
:
k
=
1
,
2
, ···,
K
}
of data matrix
A
, there exists some biclusters
B
k
=(
S
k
,F
k
)
and
B
k
=(
S
k
,F
k
)
F
k
⊆F
k
.
9. Arbitrary positioned overlapping biclusters. In the set of biclusters
such that
S
k
⊆S
k
or
B
=
{
B
k
=
(
S
k
,F
k
)
:
k
=
1
,
2
, ···,
K
}
of data matrix
A
, there exists some entry
a
ij
such that
k
. In the meantime, biclusters
B
k
,
a
ij
∈
B
k
and
a
ij
∈
B
k
with
k
=
B
k
may share
some common samples or features.
To check the nine biclustering structures, and according to above definitions of
exclusive and exhaustive, structures 1, 2 are exclusive; structure 3 is nonoverlap-
ping; structure 1 is nonexhaustive; structures 2, 3, 4, and 5 are exhaustive; and the
properties for some other structures can be found from its classification. Note that
these structures are not always strict. For example, structures 2, 3, 4, and 5 also have
nonexclusive versions (which will not satisfy above formal requirements), and for
details we refer to [37].
6.3 Biclustering Techniques and Algorithms
In this section, the biclustering techniques and algorithms are divided into several
class based on the methods used for different areas of mathematics, probability,
or other optimization methods. Here we are concentrating on mathematical back-
grounds.
6.3.1 Based on Matrix Means and Residues
For a bicluster
B
k
=(
S
k
,F
k
)
, several means based on the bicluster are defined. The
mean of row
i
of
B
k
is
1
|F
k
|
∑
j
(
r
)
ik
μ
=
a
ij
,
(6.7)
∈F
k
the mean of column
j
of
B
k
is
