<s>
Biclustering	B-Algorithm
,	O
block	O
clustering	B-Algorithm
,	O
Co-clustering	B-Algorithm
or	O
two-mode	B-Algorithm
clustering	I-Algorithm
is	O
a	O
data	B-Application
mining	I-Application
technique	O
which	O
allows	O
simultaneous	O
clustering	B-Algorithm
of	O
the	O
rows	O
and	O
columns	O
of	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
samples	O
represented	O
by	O
an	O
-dimensional	O
feature	O
vector	O
,	O
the	O
entire	O
dataset	O
can	O
be	O
represented	O
as	O
rows	O
in	O
columns	O
(	O
i.e.	O
,	O
an	O
matrix	B-Architecture
)	O
.	O
</s>
<s>
The	O
Biclustering	B-Algorithm
algorithm	O
generates	O
Biclusters	B-Algorithm
.	O
</s>
<s>
A	O
Bicluster	B-Algorithm
is	O
a	O
subset	O
of	O
rows	O
which	O
exhibit	O
similar	O
behavior	O
across	O
a	O
subset	O
of	O
columns	O
,	O
or	O
vice	O
versa	O
.	O
</s>
<s>
Biclustering	B-Algorithm
was	O
originally	O
introduced	O
by	O
John	O
A	O
.	O
Hartigan	O
in	O
1972	O
.	O
</s>
<s>
The	O
term	O
"	O
Biclustering	B-Algorithm
"	O
was	O
then	O
later	O
used	O
and	O
refined	O
by	O
Boris	O
G	O
.	O
Mirkin	O
.	O
</s>
<s>
This	O
algorithm	O
was	O
not	O
generalized	O
until	O
2000	O
,	O
when	O
Y	O
.	O
Cheng	O
and	O
George	O
M	O
.	O
Church	O
proposed	O
a	O
biclustering	B-Algorithm
algorithm	O
based	O
on	O
variance	O
and	O
applied	O
it	O
to	O
biological	O
gene	O
expression	O
data	O
.	O
</s>
<s>
In	O
2001	O
and	O
2003	O
,	O
I	O
.	O
S	O
.	O
Dhillon	O
published	O
two	O
algorithms	O
applying	O
biclustering	B-Algorithm
to	O
files	O
and	O
words	O
.	O
</s>
<s>
Dhillon	O
assumed	O
the	O
loss	O
of	O
mutual	O
information	O
during	O
biclustering	B-Algorithm
was	O
equal	O
to	O
the	O
Kullback	O
–	O
Leibler-distance	O
(	O
KL-distance	O
)	O
between	O
P	O
and	O
Q	O
.	O
P	O
represents	O
the	O
distribution	O
of	O
files	O
and	O
feature	O
words	O
before	O
Biclustering	B-Algorithm
,	O
while	O
Q	O
is	O
the	O
distribution	O
after	O
Biclustering	B-Algorithm
.	O
</s>
<s>
In	O
2004	O
,	O
Arindam	O
Banerjee	O
used	O
a	O
weighted-Bregman	O
distance	O
instead	O
of	O
KL-distance	O
to	O
design	O
a	O
Biclustering	B-Algorithm
algorithm	O
that	O
was	O
suitable	O
for	O
any	O
kind	O
of	O
matrix	B-Architecture
,	O
unlike	O
the	O
KL-distance	O
algorithm	O
.	O
</s>
<s>
The	O
complexity	O
of	O
the	O
Biclustering	B-Algorithm
problem	O
depends	O
on	O
the	O
exact	O
problem	O
formulation	O
,	O
and	O
particularly	O
on	O
the	O
merit	O
function	O
used	O
to	O
evaluate	O
the	O
quality	O
of	O
a	O
given	O
Bicluster	B-Algorithm
.	O
</s>
<s>
In	O
the	O
simple	O
case	O
that	O
there	O
is	O
an	O
only	O
element	O
a(i,j )	O
either	O
0	O
or	O
1	O
in	O
the	O
binary	O
matrix	B-Architecture
A	O
,	O
a	O
Bicluster	B-Algorithm
is	O
equal	O
to	O
a	O
biclique	O
in	O
the	O
corresponding	O
bipartite	O
graph	O
.	O
</s>
<s>
The	O
maximum	O
size	O
Bicluster	B-Algorithm
is	O
equivalent	O
to	O
the	O
maximum	O
edge	O
biclique	O
in	O
the	O
bipartite	O
graph	O
.	O
</s>
<s>
In	O
the	O
complex	O
case	O
,	O
the	O
element	O
in	O
matrix	B-Architecture
A	O
is	O
used	O
to	O
compute	O
the	O
quality	O
of	O
a	O
given	O
Bicluster	B-Algorithm
and	O
solve	O
the	O
more	O
restricted	O
version	O
of	O
the	O
problem	O
.	O
</s>
<s>
It	O
requires	O
either	O
large	O
computational	O
effort	O
or	O
the	O
use	O
of	O
lossy	O
heuristics	B-Algorithm
to	O
short-circuit	O
the	O
calculation	O
.	O
</s>
<s>
When	O
a	O
Biclustering	B-Algorithm
algorithm	O
tries	O
to	O
find	O
a	O
constant-value	O
Bicluster	B-Algorithm
,	O
it	O
reorders	O
the	O
rows	O
and	O
columns	O
of	O
the	O
matrix	B-Architecture
to	O
group	O
together	O
similar	O
rows	O
and	O
columns	O
,	O
eventually	O
grouping	O
Biclusters	B-Algorithm
with	O
similar	O
values	O
.	O
</s>
<s>
A	O
perfect	O
constant	O
Bicluster	B-Algorithm
is	O
a	O
matrix(I,J )	O
in	O
which	O
all	O
values	O
a(i,j )	O
are	O
equal	O
to	O
a	O
given	O
constant	O
μ	O
.	O
</s>
<s>
According	O
to	O
Hartigan	O
's	O
algorithm	O
,	O
by	O
splitting	O
the	O
original	O
data	O
matrix	B-Architecture
into	O
a	O
set	O
of	O
Biclusters	B-Algorithm
,	O
variance	O
is	O
used	O
to	O
compute	O
constant	O
Biclusters	B-Algorithm
.	O
</s>
<s>
Hence	O
,	O
a	O
perfect	O
Bicluster	B-Algorithm
may	O
be	O
equivalently	O
defined	O
as	O
a	O
matrix	B-Architecture
with	O
a	O
variance	O
of	O
zero	O
.	O
</s>
<s>
In	O
order	O
to	O
prevent	O
the	O
partitioning	O
of	O
the	O
data	O
matrix	B-Architecture
into	O
Biclusters	B-Algorithm
with	O
the	O
only	O
one	O
row	O
and	O
one	O
column	O
;	O
Hartigan	O
assumes	O
that	O
there	O
are	O
,	O
for	O
example	O
,	O
K	O
Biclusters	B-Algorithm
within	O
the	O
data	O
matrix	B-Architecture
.	O
</s>
<s>
When	O
the	O
data	O
matrix	B-Architecture
is	O
partitioned	O
into	O
K	O
Biclusters	B-Algorithm
,	O
the	O
algorithm	O
ends	O
.	O
</s>
<s>
Unlike	O
the	O
constant-value	O
Biclusters	B-Algorithm
,	O
these	O
types	O
of	O
Biclusters	B-Algorithm
cannot	O
be	O
evaluated	O
solely	O
based	O
on	O
the	O
variance	O
of	O
their	O
values	O
.	O
</s>
<s>
There	O
are	O
,	O
however	O
,	O
other	O
algorithms	O
,	O
without	O
the	O
normalization	O
step	O
,	O
that	O
can	O
find	O
Biclusters	B-Algorithm
which	O
have	O
rows	O
and	O
columns	O
with	O
different	O
approaches	O
.	O
</s>
<s>
For	O
Biclusters	B-Algorithm
with	O
coherent	O
values	O
on	O
rows	O
and	O
columns	O
,	O
an	O
overall	O
improvement	O
over	O
the	O
algorithms	O
for	O
Biclusters	B-Algorithm
with	O
constant	O
values	O
on	O
rows	O
or	O
on	O
columns	O
should	O
be	O
considered	O
.	O
</s>
<s>
In	O
Cheng	O
and	O
Church	O
's	O
theorem	O
,	O
a	O
Bicluster	B-Algorithm
is	O
defined	O
as	O
a	O
subset	O
of	O
rows	O
and	O
columns	O
with	O
almost	O
the	O
same	O
score	O
.	O
</s>
<s>
The	O
relationship	O
between	O
these	O
cluster	O
models	O
and	O
other	O
types	O
of	O
clustering	B-Algorithm
such	O
as	O
correlation	B-Algorithm
clustering	I-Algorithm
is	O
discussed	O
in	O
.	O
</s>
<s>
There	O
are	O
many	O
Biclustering	B-Algorithm
algorithms	O
developed	O
for	O
bioinformatics	O
,	O
including	O
:	O
block	O
clustering	B-Algorithm
,	O
CTWC	O
(	O
Coupled	O
Two-Way	O
Clustering	B-Algorithm
)	O
,	O
ITWC	O
(	O
Interrelated	O
Two-Way	O
Clustering	B-Algorithm
)	O
,	O
δ-bicluster	O
,	O
δ-pCluster	O
,	O
δ-pattern	O
,	O
FLOC	O
,	O
OPC	O
,	O
Plaid	O
Model	O
,	O
OPSMs	O
(	O
Order-preserving	O
submatrixes	O
)	O
,	O
Gibbs	O
,	O
SAMBA	O
(	O
Statistical-Algorithmic	O
Method	O
for	O
Bicluster	B-Algorithm
Analysis	O
)	O
,	O
Robust	O
Biclustering	B-Algorithm
Algorithm	O
(	O
RoBA	O
)	O
,	O
Crossing	O
Minimization	O
,	O
cMonkey	O
,	O
PRMs	O
,	O
DCC	O
,	O
LEB	O
(	O
Localize	O
and	O
Extract	O
Biclusters	B-Algorithm
)	O
,	O
QUBIC	O
(	O
QUalitative	O
BIClustering	B-Algorithm
)	O
,	O
BCCA	O
(	O
Bi-Correlation	O
Clustering	B-Algorithm
Algorithm	O
)	O
BIMAX	O
,	O
ISA	O
and	O
FABIA	O
(	O
Factor	O
analysis	O
for	O
Bicluster	B-Algorithm
Acquisition	O
)	O
,	O
runibic	O
,	O
</s>
<s>
and	O
recently	O
proposed	O
hybrid	O
method	O
EBIC	O
(	O
evolutionary-based	O
Biclustering	B-Algorithm
)	O
,	O
which	O
was	O
shown	O
to	O
detect	O
multiple	O
patterns	O
with	O
very	O
high	O
accuracy	O
.	O
</s>
<s>
More	O
recently	O
,	O
IMMD-CC	O
is	O
proposed	O
that	O
is	O
developed	O
based	O
on	O
the	O
iterative	B-Algorithm
complexity	O
reduction	O
concept	O
.	O
</s>
<s>
IMMD-CC	O
is	O
able	O
to	O
identify	O
co-cluster	B-Algorithm
centroids	O
from	O
highly	O
sparse	O
transformation	O
obtained	O
by	O
iterative	B-Algorithm
multi-mode	O
discretization	O
.	O
</s>
<s>
Biclustering	B-Algorithm
algorithms	O
have	O
also	O
been	O
proposed	O
and	O
used	O
in	O
other	O
application	O
fields	O
under	O
the	O
names	O
co-clustering	B-Algorithm
,	O
bi-dimensional	O
clustering	B-Algorithm
,	O
and	O
subspace	O
clustering	B-Algorithm
.	O
</s>
<s>
Recent	O
proposals	O
have	O
addressed	O
the	O
Biclustering	B-Algorithm
problem	O
in	O
the	O
specific	O
case	O
of	O
time-series	O
gene	O
expression	O
data	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
interesting	O
Biclusters	B-Algorithm
can	O
be	O
restricted	O
to	O
those	O
with	O
contiguous	O
columns	O
.	O
</s>
<s>
This	O
restriction	O
leads	O
to	O
a	O
tractable	O
problem	O
and	O
enables	O
the	O
development	O
of	O
efficient	O
exhaustive	O
enumeration	B-Language
algorithms	O
such	O
as	O
CCC-Biclustering	O
and	O
e-CCC-Biclustering	O
.	O
</s>
<s>
The	O
approximate	O
patterns	O
in	O
CCC-Biclustering	O
algorithms	O
allow	O
a	O
given	O
number	O
of	O
errors	O
,	O
per	O
gene	O
,	O
relatively	O
to	O
an	O
expression	O
profile	O
representing	O
the	O
expression	O
pattern	O
in	O
the	O
Bicluster	B-Algorithm
.	O
</s>
<s>
The	O
e-CCC-Biclustering	O
algorithm	O
uses	O
approximate	O
expressions	O
to	O
find	O
and	O
report	O
all	O
maximal	O
CCC-Bicluster	O
'	O
s	O
by	O
a	O
discretized	O
matrix	B-Architecture
A	O
and	O
efficient	O
string	B-Language
processing	I-Language
techniques	O
.	O
</s>
<s>
These	O
algorithms	O
find	O
and	O
report	O
all	O
maximal	O
Biclusters	B-Algorithm
with	O
coherent	O
and	O
contiguous	O
columns	O
with	O
perfect/approximate	O
expression	O
patterns	O
,	O
in	O
time	O
linear/polynomial	O
which	O
is	O
obtained	O
by	O
manipulating	O
a	O
discretized	O
version	O
of	O
original	O
expression	O
matrix	B-Architecture
in	O
the	O
size	O
of	O
the	O
time-series	O
gene	O
expression	O
matrix	B-Architecture
using	O
efficient	O
string	B-Language
processing	I-Language
techniques	O
based	O
on	O
suffix	B-Architecture
trees	I-Architecture
.	O
</s>
<s>
Some	O
recent	O
algorithms	O
have	O
attempted	O
to	O
include	O
additional	O
support	O
for	O
Biclustering	B-Algorithm
rectangular	O
matrices	O
in	O
the	O
form	O
of	O
other	O
datatypes	O
,	O
including	O
cMonkey	O
.	O
</s>
<s>
There	O
is	O
an	O
ongoing	O
debate	O
about	O
how	O
to	O
judge	O
the	O
results	O
of	O
these	O
methods	O
,	O
as	O
Biclustering	B-Algorithm
allows	O
overlap	O
between	O
clusters	O
and	O
some	O
algorithms	O
allow	O
the	O
exclusion	O
of	O
hard-to-reconcile	O
columns/conditions	O
.	O
</s>
<s>
Because	O
this	O
is	O
an	O
unsupervised	B-General_Concept
classification	I-General_Concept
problem	O
,	O
the	O
lack	O
of	O
a	O
gold	O
standard	O
makes	O
it	O
difficult	O
to	O
spot	O
errors	O
in	O
the	O
results	O
.	O
</s>
<s>
One	O
approach	O
is	O
to	O
utilize	O
multiple	O
Biclustering	B-Algorithm
algorithms	O
,	O
with	O
the	O
majority	O
or	O
super-majority	O
voting	O
amongst	O
them	O
to	O
decide	O
the	O
best	O
result	O
.	O
</s>
<s>
Another	O
way	O
is	O
to	O
analyze	O
the	O
quality	O
of	O
shifting	O
and	O
scaling	O
patterns	O
in	O
Biclusters	B-Algorithm
.	O
</s>
<s>
Biclustering	B-Algorithm
has	O
been	O
used	O
in	O
the	O
domain	O
of	O
text	B-Algorithm
mining	I-Algorithm
(	O
or	O
classification	O
)	O
which	O
is	O
popularly	O
known	O
as	O
co-clustering	B-Algorithm
.	O
</s>
<s>
Text	O
corpora	O
are	O
represented	O
in	O
a	O
vectoral	O
form	O
as	O
a	O
matrix	B-Architecture
D	O
whose	O
rows	O
denote	O
the	O
documents	O
and	O
whose	O
columns	O
denote	O
the	O
words	O
in	O
the	O
dictionary	O
.	O
</s>
<s>
Matrix	B-Architecture
elements	O
Dij	O
denote	O
occurrence	O
of	O
word	O
j	O
in	O
document	O
i	O
.	O
Co-clustering	B-Algorithm
algorithms	O
are	O
then	O
applied	O
to	O
discover	O
blocks	O
in	O
D	O
that	O
correspond	O
to	O
a	O
group	O
of	O
documents	O
(	O
rows	O
)	O
characterized	O
by	O
a	O
group	O
of	O
words(columns )	O
.	O
</s>
<s>
Text	O
clustering	B-Algorithm
can	O
solve	O
the	O
high-dimensional	O
sparse	O
problem	O
,	O
which	O
means	O
clustering	B-Algorithm
text	O
and	O
words	O
at	O
the	O
same	O
time	O
.	O
</s>
<s>
When	O
clustering	B-Algorithm
text	O
,	O
we	O
need	O
to	O
think	O
about	O
not	O
only	O
the	O
words	O
information	O
,	O
but	O
also	O
the	O
information	O
of	O
words	O
clusters	O
that	O
was	O
composed	O
by	O
words	O
.	O
</s>
<s>
This	O
is	O
called	O
co-clustering	B-Algorithm
.	O
</s>
<s>
There	O
are	O
two	O
advantages	O
of	O
co-clustering	B-Algorithm
:	O
one	O
is	O
clustering	B-Algorithm
the	O
test	O
based	O
on	O
words	O
clusters	O
can	O
extremely	O
decrease	O
the	O
dimension	O
of	O
clustering	B-Algorithm
,	O
it	O
can	O
also	O
appropriate	O
to	O
measure	O
the	O
distance	O
between	O
the	O
tests	O
.	O
</s>
<s>
This	O
corresponding	O
information	O
can	O
be	O
used	O
to	O
describe	O
the	O
type	O
of	O
texts	O
and	O
words	O
,	O
at	O
the	O
same	O
time	O
,	O
the	O
result	O
of	O
words	O
clustering	B-Algorithm
can	O
be	O
also	O
used	O
to	O
text	B-Algorithm
mining	I-Algorithm
and	O
information	O
retrieval	O
.	O
</s>
<s>
Several	O
approaches	O
have	O
been	O
proposed	O
based	O
on	O
the	O
information	O
contents	O
of	O
the	O
resulting	O
blocks	O
:	O
matrix-based	O
approaches	O
such	O
as	O
SVD	O
and	O
BVD	O
,	O
and	O
graph-based	O
approaches	O
.	O
</s>
<s>
Matrix-based	O
methods	O
focus	O
on	O
the	O
decomposition	O
of	O
matrices	O
into	O
blocks	O
such	O
that	O
the	O
error	O
between	O
the	O
original	O
matrix	B-Architecture
and	O
the	O
regenerated	O
matrices	O
from	O
the	O
decomposition	O
is	O
minimized	O
.	O
</s>
<s>
More	O
recently	O
(	O
Bisson	O
and	O
Hussain	O
)	O
have	O
proposed	O
a	O
new	O
approach	O
of	O
using	O
the	O
similarity	O
between	O
words	O
and	O
the	O
similarity	O
between	O
documents	O
to	O
co-cluster	B-Algorithm
the	O
matrix	B-Architecture
.	O
</s>
<s>
Their	O
method	O
(	O
known	O
as	O
χ-Sim	O
,	O
for	O
cross	O
similarity	O
)	O
is	O
based	O
on	O
finding	O
document-document	O
similarity	O
and	O
word-word	O
similarity	O
,	O
and	O
then	O
using	O
classical	O
clustering	B-Algorithm
methods	O
such	O
as	O
hierarchical	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
Instead	O
of	O
explicitly	O
clustering	B-Algorithm
rows	O
and	O
columns	O
alternately	O
,	O
they	O
consider	O
higher-order	O
occurrences	O
of	O
words	O
,	O
inherently	O
taking	O
into	O
account	O
the	O
documents	O
in	O
which	O
they	O
occur	O
.	O
</s>
<s>
This	O
approach	O
of	O
taking	O
higher-order	O
similarities	O
takes	O
the	O
latent	O
semantic	O
structure	O
of	O
the	O
whole	O
corpus	O
into	O
consideration	O
with	O
the	O
result	O
of	O
generating	O
a	O
better	O
clustering	B-Algorithm
of	O
the	O
documents	O
and	O
words	O
.	O
</s>
<s>
In	O
text	O
databases	O
,	O
for	O
a	O
document	O
collection	O
defined	O
by	O
a	O
document	O
by	O
term	O
D	O
matrix	B-Architecture
(	O
of	O
size	O
m	O
by	O
n	O
,	O
m	O
:	O
number	O
of	O
documents	O
,	O
n	O
:	O
number	O
of	O
terms	O
)	O
the	O
cover-coefficient	O
based	O
clustering	B-Algorithm
methodology	O
yields	O
the	O
same	O
number	O
of	O
clusters	O
both	O
for	O
documents	O
and	O
terms	O
(	O
words	O
)	O
using	O
a	O
double-stage	O
probability	O
experiment	O
.	O
</s>
<s>
The	O
generative	O
framework	O
allows	O
FABIA	O
to	O
determine	O
the	O
information	O
content	O
of	O
each	O
Bicluster	B-Algorithm
to	O
separate	O
spurious	O
Biclusters	B-Algorithm
from	O
true	O
Biclusters	B-Algorithm
.	O
</s>
