Systems, Man & Cybernetics - April 2017 - 26
then we need to find all these
The linear relations (coherence
hyperplanes and determine their
among rows or columns) in 2-D
In coclustering,
intersections. The hyperplanes can
coclusters can be extended to
be detected based on the Hough
the data is partitioned
higher-dimensional data. We pro-
transform (HT), which is widely
vide definitions for 3-D data only
in both row and
used in image processing and pat-
[25], and these definitions can be
column directions.
tern recognition [27], [28].
further extended to higher dimen-
If the input data contain a large
sions in a similar way. Assume
number of columns, then the HT-
that our input is a third-order ten-
based method that is applied to all
sor, A = ^a ijk h ! R M # M # M , and that
columns at the same time becomes ineffective because the
a 3-D cocluster, A IJK, contains elements in locations
accumulator array for the quantized hyperplane parameters
I # J # K = " i 1, i 2, g, i N , # " j 1, j 2, g, j N , # " k 1, k 2, g, k N , .
is too large. The problem can be overcome if we only consid-
We define the following 3-D coclusters:
er two columns at a time. Subcoclusters detected from col-
1) Constant cocluster: A IJK = " a ijk = n | i ! I, j ! J, k ! K ,
umn-pair spaces can then be merged to form larger ones.
2) Constant-column-fiber cocluster:
For 3-D and higher-dimensional data, there will be a
A IJK = " a Ijk = n jk 1 I | j ! J, k ! K ,
large number of column pairs. This problem can be solved
3) Additive-fiber cocluster:
if we transform the input data first using singular-value
a
A IJK = " a I jk = n jk 1 I + a I(1) ; j ! J, k ! K ,, where I(1) is the
decomposition (SVD) [25], which will be discussed in the
first fiber of the co-cluster
next section.
4) Multiplicative-fiber cocluster:
A IJK = " a Ijk = n jk a I(1) ; j ! J, k ! K , .
Cocluster Detection in Singular-Vector Spaces
In practical applications, a cocluster pattern often contains
In this section, we describe a coclustering method based on
noise, and the linear relations defined above only hold approxi-
hyperplane detection in singular-vector spaces (HDSVS),
mately. Also, there can be more than one cocluster in the data,
which we have recently developed [25]. This method can be
and we need to determine the index sets I and J for 2-D data or
used to attend to all of the types of coclusters previously dis-
I, J, and K for 3-D data for each cocluster.
cussed and low-rank matrices or tensors in general that are
embedded in high-dimensional data arrays.
Coclustering Algorithms
Let us consider 2-D coclusters first. According to their
Cheng and Church [1] were the first who introduced cocluster-
definition given previously, an additive cocluster as a matrix
ing to gene expression data analysis. In their method, the fol-
has rank two, and constant, constant-row, constant-column,
lowing cost function is minimized to find a 2-D cocluster
and multiplicative coclusters have rank one. Applying SVD
defined on I # J:
to AIJ, we obtain
2
1 ||
^a - ar iJ - ar Ij + ar ij h ,
C (I, J) =
(1)
I J i ! I j ! J ij
v 1 0 ^ v 1 hT
E<
F
A IJ = URV T = 6u 1 u 2@;
0 v 2 ^ v 2 hT
v 1 0 v 11 v 12 g v 1M
where ar iJ = ^1/ J h| j ! J a ij, ar Ij = ^1/ I h| i ! I a ij, and ar IJ =
E;
E
= 6u 1 u 2@;
0 v 2 v 21 v 22 g v 2M
^ 1/ I J h| i ! I | j ! J a ij represent row average, column aver-
= 6v 1 v 11 u 1 + v 2 v 21 u 2 v 1 v 12 u 1
age, and pattern average, respectively, of the cocluster. The
+ v 2 v 22 u 2 g v 1 v 1M u 1 + v 2 v 2M u 2@,
method can be used to detect constant coclusters. It is inter-
(2)
esting to note that in (1), if we remove the row and column
averages and change the sign of the pattern average, then the
where U, R , and V are matrices containing left singular vec-
cost function looks similar to that used for the k-means clus-
tors, singular values, and right singular vectors, respectively.
tering algorithm.
Here, we only keep two leading singular values because the
A number of other coclustering methods have also been
rank of A IJ is at most two.
developed recently [2]-[26]. Most of these methods can be
Equation (2) shows that each column a IJ of A IJ can be
applied to 2-D data and deal with a specific type of coclusters
expressed as a linear combination of singular vectors u 1
only. Several review papers provide detailed summaries and
and u 2 :
comparisons of these methods [3], [16], [20].
Our research group has proposed geometric models for
v 1 v 1j u 1 + v 2 v 2j u 2 = a Ij .
(3)
coclusters [6], [7], [9], [15], [19], [25]. According to the defi-
nitions of 2-D coclusters discussed previously, columns of
This set of linear equations represents a line in the u 1 and
a cocluster can be related by linear equations. Geometri-
u 2 space, while elements of u 1 and u 2 are points on the line.
cally, these equations correspond to lines, planes, or
Similarly, we can show that each row of A IJ can be expressed
hyperplanes in 2-D, 3-D, or higher-dimensional spaces,
as a linear combination of singular vectors, v 1 and v 2, and ele-
respectively. If we consider all columns at the same time,
ments of v 1 and v 2 form a line.
1
2
I
3
J
K
2
2
2
26
IEEE SyStEmS, man, & CybErnEtICS magazInE A pri l 2017
2
Table of Contents for the Digital Edition of Systems, Man & Cybernetics - April 2017
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