社団法人 電子情報通信学会
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS

信学技報
TECHNICAL REPORT OF IEICE.

Batch-Learning Self-Organizing Map with False-Neighbor Degree
for Eﬀective Self-Organization
Haruna MATSUSHITA† and Yoshifumi NISHIO†† Department of Electrical and Electronic Engineering, Tokushima University
2–1 Minami-Josanjima, Tokushima 770–8506, Japan
E-mail: †{haruna,nishio}@ee.tokushima-u.ac.jp
Abstract This study proposes a Batch-Learning Self-Organizing Map with False-Neighbor degree between neurons (called BL-FNSOM). False-neighbor degrees are allocated between adjacent rows and adjacent columns of
BL-FNSOM. The initial values of all of the false-neighbor degrees are set to zero, however, they are increased withlearning, and the false-neighbor degrees act as a burden of the distance between map nodes when the weight vectorsof neurons are updated. BL-FNSOM changes the neighborhood relationship more ﬂexibly according to the situation and the shape of data although using batch learning. We apply BL-FNSOM to some input data and conﬁrm
that FN-SOM can obtain a more eﬀective map reﬂecting the distribution state of input data than the conventional
Batch-Learning SOM.
Key words self-organizing maps (SOM), clustering, unsupervised learning

1. Introduction
Since we can accumulate a huge amount of data in recent
years, it is important to investigate various clustering methods [1]. The Self-Organizing Map (SOM) is an unsupervisedneural network [2] and has attracted attention for its clustering properties. In the learning algorithm of SOM, a winner,which is a neuron closest to the input data, and its neighboring neurons are updated, regardless of the distance between
the input data and the neighboring neurons. For this reason, if we apply SOM to clustering of the input data which
includes some clusters located at distant locations, there aresome inactive neurons between clusters where without the
input data. Because inactive neurons are on a part without
the input data, we are misled into thinking that there aresome input data between clusters.
Then, what are the “neighbors”? In the real world, it is notalways true that the next-door house is close to my house.
In other words, “neighbors” are not always “true neighbors”.
In addition, the relationship between neighborhoods is not
ﬁxed, but keeps changing with time. It is important tochange the neighborhood relationship ﬂexibly according to
the situation.
On the other side, the synaptic strength is not constant in
the brain. So far, the Growing Grid network was proposed
in 1985 [3]. Growing Grid increases the neighborhood distance between neurons by increasing the number of neurons.
However, there is not much research changing the synaptic
strength even though there are algorithms which increase thenumber of neurons or consider rival neurons [4].
In our past study, we proposed the SOM with False-

Neighbor degree between neurons (called FN-SOM) [5]. The
false-neighbor degrees act as a burden of the distance between map nodes when the weight vectors of neurons are
updated. We have conﬁrmed that there are few inactive
neurons using FN-SOM, and FN-SOM can obtain the mosteﬀective map reﬂecting the distribution state of input data.
Meanwhile, there are two well-known learning algorithms
for SOM: sequential learning and batch learning. In the sequential learning, the winner for an input data is found andthe weight vectors are updated immediately. However, thesequential learning has some problems as a large calculation amount and a dependence on order of the input data.
In the batch learning, the updates are deferred to the endof a leaning, namely the presentation of the whole dataset.
Batch-Learning SOM (BL-SOM) [2] is used to speed up thecalculating time and to remove the dependence on order ofthe input data.
In this study, we propose a Batch-Learning Self-Organizing
Map with False-Neighbor degree between neurons (called
BL-FNSOM). We improve the previously proposed FN-SOM
and apply the batch learning to FN-SOM. The initial values of all of the false-neighbor degrees are set to zero,
however, they are increased with learning, and the falseneighbor degrees act as a burden of the distance between map
nodes when the weight vectors of neurons are updated. BLFNSOM changes the neighborhood relationship more ﬂexibly according to the situation and the shape of data althoughusing batch learning.
We explain the learning algorithm of BL-FNSOM in detail in Section 3. We apply BL-FNSOM to a uniform input data to investigate an eﬀect of the false-neighbor degree.
The learning behaviors of BL-FNSOM for 2-dimensional in—1—

put data and 3-dimensional data, which have some clustering
problem, are investigated in Section 4. In addition, we apply
BL-FNSOM to a real world data set, Iris data. Learning performance is evaluated both visually and quantitatively using
three measurements and is compared with the conventional
BL-SOM. We conﬁrm that FN-SOM can obtain the more
eﬀective map reﬂecting the distribution state of input data
than BL-SOM.

C1(BLSOM1) An input vector xj is inputted to all the neurons at the same time in parallel.
(BLSOM2) Distances between xj and all the weight vectors are calculated. The input vector xj belongs to a winner
neuron which is with the weight vector closest to xj .
cj = arg min{kwi − xj k},(1)idenotes the index of the winner of the input vector xj , wherek · k is the distance measure, Euclidean distance.
(BLSOM3) After repeating the steps (BLSOM1) and (BLSOM2) for all the input data set, all the weight vectors are
updated as

∑N
wnew
=
i

j=1

∑N

hcj ,i xj

,

(2)

h
j=1 cj ,iwhere t denotes a training step and hcj ,i is the neighborhood
function around the winner cj :(hcj ,i = exp −kr i − r cj k2
2σ 2 (t)),(3)

where r i and r cj are coordinates of the i-th and the winner cunit on competitive layer, namely, kr i − r cj k is the distance
between map nodes cj and i on the map grid. σ(t) decreases
with time, in this study, we use following equation;
σ(t) = σ0 (1 − t/tmax ),(4)

where σ0 is the initial value of σ, and tmax is the maximum
number of the learning.
(BLSOM4) The steps from (BLSOM1) to (BLSOM3) sufﬁcient time.

Cm-1

R1
R2

n

Rn-1

2. Batch-Learning Self-Organizing Map(BL-SOM)
We explain a batch learning algorithm of SOM in detail.
SOM consists of n × m neurons located at 2-dimensional
rectangular grid. Each neuron i is represented by a ddimensional weight vector wi = (wi1 , wi2 , · · · , wid ) (i =1, 2, · · · , nm), where d is equal to the dimension of the input
vector xj = (xj1 , xj2 , · · · , xjd ) (j = 1, 2, · · · , N ). Also batchlearning algorithm is iterative, but instead of using a single data vector at a time, the whole data set is presented tothe map before any adjustments are made. The initial weightvectors of BL-SOM are given at orderly position based on theﬁrst and second principal components of the d-dimensional
space by the Principal Component Analysis (PCA) [6].

C2mFig. 1 A false-neighbor degree of row Rr (1 <= r <= n − 1) andcolumn Ck (1 <= k <= m − 1). Neurons of FN-SOM arelocated at a n × m rectangular grid.

3. BL-SOM with False-Neighbor Degree(BL-FNSOM)
We explain a Batch Learning SOM with False-Neighbor
Degree between neurons (BL-FNSOM) in detail. Falseneighbor degrees of rows Rr (1 <=r <= n − 1) are allocatedbetween adjacent rows of BL-FNSOM with the size of n × m
grid (as Fig. 1).
Likewise, false-neighbor degrees of columns Ck (1 <=k <=m − 1) are allocated between adjacent columns of BLFNSOM. In other words, R1 means the false-neighbor degree between neurons of the 1st row and the 2nd row, andC4 is the false-neighbor degree between neurons of the 4thcolumn and the 5th column. The initial values of the falseneighbor degrees are set to zero. and the initial values of all
the weight vectors are given over the input space at random.
Moreover, a winning frequency γi is associated with eachneuron and is set to zero initially. The initial weight vectorsof BL-FNSOM are given as BL-SOM method. Learning algorithm of BL-FNSOM contains two steps: Learning steps
and Considering False-Neighbors.
Learning Steps(BL-FNSOM1) An input vector xj is inputted to all theneurons at the same time in parallel.
(BL-FNSOM2) Distances between xj and all the weightvectors are calculated, and winner cj is chosen according to
Eq. (1).
(BL-FNSOM3) The winning frequency of winner cj is increased byγcj new = γcj old + 1.

(5)

(BL-FNSOM4) After repeating the steps (BL-FNSOM1)
and (BL-FNSOM3) for all the input data set, all the weightvectors are updated as

∑N
wnew
=
i

j=1

∑N

hF cj ,i xj

j=1

,

(6)

hF cj ,iwhere hF cj ,i is the neighborhood function of FN-SOM:—2—

C1=2 C2=1

(1,1)

C2

R1=1

(2,3)

q

fq

fq
R3

q

(a)
(a)

(b)

Fig. 2 Neighborhood distance between neuron 1 and neuron
7. (a) Conventional SOM method according to Eq. (3).
(b) FN-SOM method according to Eq. (8).

(hF cj ,i = exp −dF (i, cj )2σ 2 (t)

)
.

(
+

∑

)2

r2 −1

|r1 − r2 | +dF (s1 , s2 ) =Fig. 3 Increment of the false-neighbor degree. (a) q and its falseneighbor fq are in the 3rd row and in the 2nd and 3rdcolumn, respectively. Then, the false-neighbor degree C2
between columns 2 and 3 is increased by Eq. (11). (b) q
and fq are in the 2nd column and in the 4th and 3rd row,
respectively. The false-neighbor degree R3 between rows
3 and 4 is increased by Eq. (12).

(7)

dF (i, cj ) is the neighboring distances between the winner cj
and the other neurons calculated with considering the falseneighbor degrees. For instance, for two neurons s1 , which
is located at r1 -th row and k1 -th column, and s2 , which islocated at r2 -th row and k2 -th column, the neighboring distance is deﬁned as the following measure;

(

(b)

Rr

)2

r=r1
k2 −1

|k1 − k2 | +

∑

Ck

(8)
,

k=k1

where r1 < r2 , k1 < k2 , namely, r=r1 Rr means the sum of
the false-neighbor degrees between the rows r1 and r2 , and∑k2 −1
Ck means the sum of the false-neighbor degrees bek=k1
tween the column k1 and k2 (as Fig. 2).
∑nm
γ > λ is satisﬁed, we ﬁnd the(BL-FNSOM5) If
i=1 i =
false-neighbors and increase the false-neighboring degree, according to steps from (FN-SOM6) to (FN-SOM9). If not, we
perform step (FN-SOM10). In other words, we consider thefalse-neighbors every time when the learning steps are performed for λ input data.
Considering F alse-N eighbors(BL-FNSOM6) We ﬁnd a set of neurons S which havenever become the winner:(9)

If the neuron, which have never become the winner, does notexist, namely S = ∅, we return to (FN-SOM1) without considering the false-neighbors.
(BL-FNSOM7) A false-neighbor fq of each neuron q in S
is chosen from the set of direct topological neighbors of q denoted as Nq 1 . fq is the neuron whose weight vector is most
distant from q;
fq = arg max{kwi − wq k}, q ∈ S, i ∈ Nq 1 .

Ck new = Ck old +

(10)

n+m
.
2nm

(11)

In the same way, if q and fq are in the k-th column and in the(r + 1)-th and r-th row (as Fig. 3(b)), the false-neighbor degree Rr between rows r and r + 1 is also increased accordingto
Rr new = Rr old +∑r2 −1S = {i | γi = 0}.

(BL-FNSOM8) A false-neighbor degree between each q
and its false-neighbor fq , Rr or Ck , is increased. If q and fq
are in the r-th row and in the k-th and (k + 1)-th column (asFig. 3(a)), the false-neighbor degree Ck between columns k
and k + 1 is increased according to

n+m
.
2nm

(12)

These amounts of increasing the false-neighbor degree arederived by the number of neurons numerically and are ﬁxed.
(BL-FNSOM9) The winning frequency of all the neurons
are reset to zero: γi = 0.
(BL-FNSOM10) The steps from (FN-SOM1) to (FNSOM9) are repeated for all the input data.

4. Experimental Results
We apply BL-FNSOM to various input data and compare
BL-FNSOM with the conventional BL-SOM.
4. 1 For 2-dimensional data
First, we consider 2-dimensional input data as shown in
Fig. 4(a). The input data is Target data set, which has aclustering problem of outliers [7]. The total number of the
input data N is 770, and the input data has six clusters
which include 4 outliers.
Both BL-SOM and BL-FNSOM have nm = 100 neurons(10 × 10). We repeat the learning 20 times for all input data,
namely tmax = 15400. The parameters of the learning arechosen as follows;(For SOM)

σ0 = 4(For FN-SOM)

σ0 = 4, λ = 3000,

iwhere we use the same σ0 to BL-SOM and BL-FNSOM for
—3—

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(c)

Fig. 4 Learning results for Target data. (a) Input data and initial state. (b) Conventional
BL-SOM. (c) BL-FNSOM.

the comparison and the conﬁrmation of the false-neighbor
degree eﬀect.
The learning result of the conventional BL-SOM is shownin Fig. 4(b). We can see that there are some inactive neurons
between clusters. The other side, the result of BL-FNSOM
is shown in Fig. 4(c). We can see from this ﬁgure that there
are just a few inactive neurons between clusters, and BLFNSOM can obtain the more eﬀective map reﬂecting the
distribution state of input data than BL-SOM.
Furthermore, we use the following three measurements toevaluate the training performance of the two algorithms.
Quantization Error Qe: This measures the average distance between each input vector and its winner [2];1 ∑kxj − w̄j k,
N
N

Qe =

(13)

j=1

where w̄j is the weight vector of the corresponding winner ofthe input vector xj . Therefore, the small value Qe is moredesirable.
Topographic Error Te: This describes how well the SOM
preserves the topology of the studied data set [8];1 ∑u(xj ),
N
N

Te =

(14)

j=1

where N is the total number of input data, u(xj ) is 1 if
the winner and 2nd winner of xj are NOT 1-neighbors eachother, otherwise u(xj ) is 0. The small value T e is more desirable. Unlike the quantization error, it considers the structureof the map. For a strangely twisted map, the topographic
error is big even if the quantization error is small.
Neuron Utilization U: This measures the percentage ofneurons that are the winner of one or more input vector in
the map [9];
1 ∑
ui ,
nm
nm

U=

(15)

i=1

where ui = 1 if the neuron i is the winner of one or more
input data. Otherwise, ui = 0. Thus, U nearer 1.0 is moredesirable.

Table 1 Quantization error Qe, Topographic error T e and Neuron utilization U for Target data.
BL-SOM
BL-FNSOM
Improvement rate

Qe
0.01440.01319.02%

Te0.2156
0.192210.85%

U
0.82
0.98
19.51%

The calculated three measurements are shown in Table 1.
The quantization error Qe of BL-FNSOM is smaller value
than BL-SOM, and by using BL-FNSOM, the quantization error Qe has improved 9.02% from using the conventional BL-SOM. This is because the result of BL-FNSOM
has few inactive neurons, therefore, the more neurons canself-organize the input data. This is conﬁrmed by the neuron utilization U . The neuron utilization U of BL-FNSOM islarger value than BL-SOM. It means that 98% neurons of BLFNSOM are the winner of one or more input data, namely,
there are few inactive neurons. On the other hand, the topographic error T e of BL-FNSOM is smaller value although
Qe and U are better values. It means that BL-FNSOM selforganizes most eﬀectively with maintenance of top quality
topology.
4. 2 For 3-dimensional data
Next, we consider 3-dimensional input data.
4. 2. 1 Atom data
We consider Atom data set shown in Fig. 5(a), which hasclustering problems of linear not separable, diﬀerent densities and variances. The total number of the input data N is
800, and the input data has two clusters.
We repeat the learning 19 times for all input data, namely
tmax = 15200. The learning conditions are the same used inFig. 4 except σ0 = 5.
The learning results of the conventional BL-SOM and BLFNSOM are shown in Figs. 5(b) and (c). We can see that
BL-FNSOM can self-organize edge data more eﬀective than
BL-SOM.
The calculated three measurements are shown in Table. 2.
In all measurements, FN-BLSOM can obtain better value
than BL-SOM.
4. 2. 2 Hepta data
We consider Hepta data set shown in Fig. 6(a), which has
a clustering problem of diﬀerent densities in clusters. The
total number of the input data N is 212, and the input data

—4—

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Fig. 5 Learning results for Atom data. (a) Input data and initial state. (b) Conventional
BL-SOM. (c) BL-FNSOM.

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Fig. 6 Learning results for Hepta data. (a) Input data and initial state. (b) Conventional
BL-SOM. (c) BL-FNSOM.

Table 2 Quantization error Qe, Topographic error T e and Neuron utilization U for Atom data.
Qe
TeU
BL-SOM
0.05070.29750.84
BL-FNSOM
0.04610.2713
0.93
Improvement rate 9.07%
8.81% 10.71%

input data of cluster part In other words, we can obtain moredetail on the cluster input data in the high dimensional data.
0.3

0.50

0.25

0.40

0.2

Table 3 Quantization error Qe, Topographic error T e and Neuron utilization U for Hepta data.
BL-SOM
BL-FNSOM
Improvement rate

Qe
0.02640.021120.08%

Te
0.32080.268916.18%

U
0.72
0.97
34.72%

0.30

0.15
0.20
0.1
0.10
0.05
0

(a)

(b)

Fig. 7 U-matrix of learning results for Hepta data. (a) Conventional BL-SOM. (b) BL-FNSOM.

has seven clusters.
We repeat the learning 70 times for all the input data,
namely tmax = 14840. The learning conditions are the sameused in Fig. 5.
Figures 6(b) and (c) show the learning results of BL-SOM
and BL-FNSOM, respectively. We can see that BL-FNSOM
has fewer inactive neurons than BL-SOM between clusters.
Table 3 shows the calculated three measurements. All
measurements are improved signiﬁcantly from using BLSOM. Because the inactive neurons have been reduced by
34.72%, more neurons are attracted to clusters and Qe has
been decreased by about 20%. It means that BL-FNSOM
can extract the feature of the input data more eﬀectively.
In order to investigate extracted cluster structure using
each algorithm, we calculate U-matrix [10] of the learning
results. U-matrix can visualize distance relationships in ahigh dimensional data space. Figure 7 shows the U-matrix
from Figs. 6(b) and (c). From these ﬁgures, we can see thatcluster boundaries of BL-FNSOM are clearer than BL-SOM.
It means that more neurons of BL-FNSOM self-organize the

4. 3 Real world data
Furthermore, we apply BL-FNSOM to the real world clustering problem. We use the Iris plant data [11] as real data.
This data is one of the best known databases to be found in
the pattern recognition literature [12]. The data set contains
three clusters of 50 instances respectively, where each classrefers to a type of iris plant. The number of attributes is four
as the sepal length, the sepal width, the petal length and thepetal width, namely, the input data are 4-dimension. The
three classes correspond to Iris setosa, Iris versicolor andIris virginica, respectively. Iris setosa is linearly separablefrom the other two, however Iris versicolor and Iris virginica
are not linearly separable from each other.
We repeat the learning 100 times for all input data, namely
tmax = 15000. The input data are normalized and are sorted
at random. The learning conditions are the same used inFig. 5.
Table 4 shows the calculated three measurements. We canconﬁrm that all measurement are improved signiﬁcantly from—5—

Table 4 Quantization error Qe, Topographic error T e and Neuron utilization U for Iris data.
Qe
TeU
BL-SOM
0.01500.37330.82
BL-FNSOM
0.01370.2867
0.91
Improvement rate 8.67% 23.20% 10.98%
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Fig. 8 U-matrix of learning results for Iris data. Label 1, 2 and 3
correspond to Iris setosa, Iris versicolor and Iris virginica,
respectively. (a) Conventional BL-SOM. (b) BL-FNSOM.

using BL-SOM. We can say that BL-FNSOM has fewer inactive neurons than BL-SOM and can self-organize more effective with keeping good map topographic.
The U-matrix calculated from each learning result are
shown in Fig. 8. The boundary line of BL-FNSOM between
Iris setosa and the other two is clearer than the conventional
BL-SOM. Meanwhile, it is hard to recognize the boundaryline between Iris versicolor and Iris virginica on both results.
However, on the BL-FNSOM result, we can faintly observe
the boundary line. Because BL-FNSOM has few inactive
neurons, more neurons can self-organize the cluster input
data, and BL-FNSOM can extract slight diﬀerent between
Iris versicolor and Iris virginica.

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5. Conclusions
In this study, we have proposed the Batch-Learning SelfOrganizing Map with False-Neighbor degree between neurons (BL-FNSOM). We have applied BL-FNSOM to 2dimensional data, 3-dimensional data and the real world
data. Learning performance has evaluated both visually andquantitatively and has compared with BL-SOM. We have
conﬁrmed that BL-FNSOM has fewer inactive neurons andcan obtain the more eﬀective map reﬂecting the distribution
state of input data than BL-SOM.

Acknowledgment
This work was partly supported by Yamaha Music Foundation.
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