Batch-Learning Self-Organizing Map
with False-Neighbor Degree between Neurons
Haruna Matsushita, Student Member, IEEE and Yoshifumi Nishio, Senior Member, IEEE

Abstract— This study proposes a Batch-Learning SelfOrganizing 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 tozero, 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 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 theconventional Batch-Learning SOM.

I. I NTRODUCTION
INCE 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 unsupervised neural 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 thisreason, 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 not always true that neighboring houses are physicallyadjacent or close to each other. In other words, “neighbors”
are not always “true neighbors”. In addition, the relationshipbetween neighborhoods is not fixed, but keeps changing withtime. It is important to change the neighborhood relationshipflexibly 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 in1985 [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 FalseNeighbor degree between neurons (called FN-SOM) [5].

S

Haruna Matsushita and Yoshifumi Nishio are with the Department ofElectrical and Electronic Engineering, Tokushima University, Tokushima,
770–8506, Japan (phone: +81–88–656–7470; fax: +81–88–656–7471; email:{haruna, nishio}@ee.tokushima-u.ac.jp).

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 thesequential 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 end of
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 SelfOrganizing 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, andthe false-neighbor degrees act as a burden of the distance
between map nodes when the weight vectors of neurons are
updated. BL-FNSOM changes the neighborhood relationshipmore flexibly according to the situation and the shape of data
although using batch learning.
We explain the learning algorithm of BL-FNSOM in detailin Section III. 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 input
data and 3-dimensional data, which have some clustering
problem, are investigated in Section IV. 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 amore effective map reflecting the distribution state of input
data than BL-SOM.
II. BATCH -L EARNING S ELF -O RGANIZING M AP
(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 ). Also2260c
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batch learning algorithm is iterative, but instead of using asingle 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].

R1(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 .

Rn-1

cj = arg min{wi − xj },i(1)denotes the index of the winner of the input vector xj , where ·  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
j=1 hcj ,i xj
=
,
(2)
w new
N
i
j=1 hcj ,iwhere t denotes a training step and hcj ,i is the neighborhood
function around the winner cj :ri − r cj 2
hcj ,i = exp −,(3)
2σ 2 (t)where r i and r cj are coordinates of the i-th and the winner cunit on competitive layer, namely, ri − r cj  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.
III. BL-SOM WITH FALSE -N EIGHBOR D EGREE(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 adjacentcolumns of BL-FNSOM. In other words, R1 means thefalse-neighbor degree between neurons of the 1st row andthe 2nd row, and C4 is the false-neighbor degree between
neurons of the 4th column and the 5th column. The initialvalues of the false-neighbor degrees are set to zero. and theinitial values of all the weight vectors are given over the
input space at random. Moreover, a winning frequency γi
is associated with each neuron and is set to zero initially.
The initial weight vectors of BL-FNSOM are given as BLSOM method. Learning algorithm of BL-FNSOM contains
two steps: Learning steps and Considering False-Neighbors.

C2

Cm-1

R2
n

m
Fig. 1. A false-neighbor degree of row Rr (1 ≤ r ≤ n − 1) and column
Ck (1 ≤ k ≤ m − 1). Neurons of FN-SOM are located at a n × m
rectangular grid.

A flowchart of the learning algorithm is shown in Fig. 2.
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 by(5)
γcj new = γcj old + 1.
(BL-FNSOM4) After repeating the steps (BL-FNSOM1) and(BL-FNSOM3) for all the input data set, all the weightvectors are updated as
N
j=1 hF cj ,i xj
,
(6)
=
w new
N
i
j=1 hF cj ,iwhere hF cj ,i is the neighborhood function of FN-SOM:
dF (i, cj )hF cj ,i = exp −.
(7)
2σ 2 (t)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
is located at r2 -th row and k2 -th column, the neighboring
distance is defined as the following measure;

2
r
2 −1
|r1 − r2 | +RrdF (s1 , s2 ) =


r=r1
k
2 −1

+ |k1 − k2 | +

2
Ck

(8)
,

k=k1

r2 −1
where r1 < r2 , k1 < k2 , namely, r=r
Rr means the sum
1of thefalse-neighbor
degrees
betweentherows r1 and r2 ,
 2 −1and kk=k
Ck means the sum of the false-neighbor degrees
1
between the column k1 and k2 (as Fig. 3).

2008 International Joint Conference on Neural Networks (IJCNN 2008)

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StartInput
Find winner

Learning stepYes
No
Update weight vectors

No
Yes
Find
No

Considering
False-Neighbors

Yes
Find false-neighbor
Increase false-neighbor degree
between each
and
Reset

Yes
No
End

Fig. 2.

Flowchart of BL-FNSOM.

nm(BL-FNSOM5) If
i=1 γi ≥ λ is satisfied, we find thefalse-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
the false-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 have never
become the winner:
S = {i | γi = 0}.

(9)

If the neuron, which have never become the winner, does
not exist, namely S = ∅, we return to (FN-SOM1) without

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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 ismost distant from q;
fq = arg max{wi − wq }, q ∈ S, i ∈ Nq 1 .
i(10)(BL-FNSOM8) A false-neighbor degree between each q andits 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. 4(a)), the false-neighbor degree Ck between columns k2008 International Joint Conference on Neural Networks (IJCNN 2008)

C1=2 C2=1

(1,1)

A. For uniform data
First, in order to confirm that the false-neighbor degree hasa bad effect or not, we consider a 2-dimensional input data,
which is uniformity and has no cluster. The total number ofthe input data N is 1000. We repeat the learning 15 times
for all input data. The parameters of the learning are chosen
as follows;(For SOM)
σ0 = 4.5,

R1=1

(2,3)

(For FN-SOM)(a)σ0 = 4.5, λ = 3000,(b)

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

C2

q

fq

fq

where we use the same σ0 to BL-SOM and BL-FNSOM forthe comparison and the confirmation of the false-neighbor
degree effect.
Learning results of the two algorithms are shown in Fig. 5.
We can see that the result of BL-FNSOM is completely sameas the conventional 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];

R3
q

(a)

1 
xj − w̄j ,
N j=1
N

Qe =
(b)

Fig. 4. Increment of the false-neighbor degree. (a) q and its false-neighbor
fq are in the 3rd row and in the 2nd and 3rd column, respectively. Then, thefalse-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).

(13)

where w̄j is the weight vector of the corresponding winnerof the 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 j=1
N

Te =

and k + 1 is increased according to
Ck new = Ck old +

n+m
.
2nm

(11)

In the same way, if q and fq are in the k-th column and inthe (r + 1)-th and r-th row (as Fig. 4(b)), the false-neighbor
degree Rr between rows r and r + 1 is also increased
according to
Rr new = Rr old +

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 (FN-SOM9)
are repeated for all the input data.
IV. E XPERIMENTAL R ESULTS
We apply BL-FNSOM to various input data and compare
BL-FNSOM with the conventional BL-SOM.

(14)

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
each other, otherwise u(xj ) is 0. The small value T e ismore desirable. Unlike the quantization error, it considers
the structure of the map. For a strangely twisted map, thetopographic error is big even if the quantization error issmall.
Neuron Utilization U: This measures the percentage ofneurons that are the winner of one or more input vector in
the map [9];
nm
1 
ui ,
(15)
U=
nm 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.
The calculated three measurements are shown in Table. I.
We can confirm that the all measurements of BL-FNSOM
are same as the conventional BL-SOM. This is because
all neurons become winner, namely false-neighbors are not2008 International Joint Conference on Neural Networks (IJCNN 2008)

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

TABLE ITABLE IIQ UANTIZATION ERROR Qe, T OPOGRAPHIC ERROR T e AND N EURON

Q UANTIZATION ERROR Qe, T OPOGRAPHIC ERROR T e AND N EURONUTILIZATION U FOR NO CLUSTER DATA .

UTILIZATION U FOR TARGET DATA .

BL-SOM
BL-FNSOM

Qe
0.03490.0349

Te
0.038
0.038

U
1.0
1.0

chosen, for the uniform input data. We can say that the falseneighbor degree does not have a bad effect.
B. For 2-dimensional data
Next, we consider 2-dimensional input data as shown in
Fig. 6(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 learning conditions are the sameused in Fig. 5 except σ0 = 5.
The learning result of the conventional BL-SOM is shownin Fig. 6(b). We can see that there are some inactive neurons
between clusters. The other side, the result of BL-FNSOM
is shown in Fig. 6(c). We can see from this figure that
there are just a few inactive neurons between clusters, and
BL-FNSOM can obtain more effective map reflecting the2264

BL-SOM
BL-FNSOM
Improvement rate

Qe
0.01440.01319.02%

Te0.2156
0.192210.85%

U
0.82
0.98
19.51%

distribution state of input data than BL-SOM.
The calculated three measurements are shown in Table II.
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, more neurons can self-organize
the input data. This is confirmed by the neuron utilization U .
The neuron utilization U of BL-FNSOM is larger value than
BL-SOM. It means that 98% neurons of BL-FNSOM 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 self-organizes mosteffectively with maintenance of top quality topology.
C. For 3-dimensional data
Next, we consider 3-dimensional input data.

2008 International Joint Conference on Neural Networks (IJCNN 2008)

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TABLE IIIQ UANTIZATION ERROR Qe, T OPOGRAPHIC ERROR T e AND N EURONTABLE IVQ UANTIZATION ERROR Qe, T OPOGRAPHIC ERROR T e AND N EURONUTILIZATION U FOR ATOM DATA .

UTILIZATION U FOR H EPTA DATA .

BL-SOM
BL-FNSOM
Improvement rate

Qe
0.05070.04619.07%

Te
0.29750.2713
8.81%

U
0.84
0.93
10.71%

1) Atom data: We consider Atom data set shown in
Fig. 7(a), which has clustering problems of linear not separable, different densities and variances. The total number ofthe 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. 5 except σ0 = 5.
The learning results of the conventional BL-SOM and BLFNSOM are shown in Figs. 7(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. III.
In all measurements, FN-BLSOM can obtain better value
than BL-SOM.
2) Hepta data: We consider Hepta data set shown in
Fig. 8(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 has seven clusters.
We repeat the learning 70 times for all the input data,
namely tmax = 14840. The learning conditions are the same

BL-SOM
BL-FNSOM
Improvement rate

Qe
0.02640.021120.08%

Te
0.32080.268916.18%

U
0.72
0.97
34.72%

used in Fig. 7.
Figures 8(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 IV 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 9 shows the U-matrix
from Figs. 8(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 theinput data of cluster part In other words, we can obtain moredetail on the cluster input data in the high dimensional data.

2008 International Joint Conference on Neural Networks (IJCNN 2008)

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0.3

TABLE VQ UANTIZATION ERROR Qe, T OPOGRAPHIC ERROR T e AND N EURONUTILIZATION U FOR I RIS DATA .

0.25

0.2

0.15

Qe
0.01500.0137
8.67%

BL-SOM
BL-FNSOM
Improvement rate

Te0.37330.286723.20%

U
0.82
0.91
10.98%

0.1

0.05

(a)
0.50

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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Fig. 9. U-matrix of learning results for Hepta data. (a) Conventional BLSOM. (b) BL-FNSOM.

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D. Real world data
2

Furthermore, we apply BL-FNSOM to the real world
clustering problem. We use the Iris plant data [11] as realdata. This data is one of the best known databased to be foundin 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 andthe petal width, namely, the input data are 4-dimension. The
three classes correspond to Iris setosa, Iris versicolor and Irisvirginica, respectively. Iris setosa is linearly separable fromthe 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. 7.
Table V shows the calculated three measurements. We
can confirm that all measurement are improved significantly
from 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. 10. 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.

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(b)
Fig. 10.
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.

V. C ONCLUSIONSIn 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 haveconfirmed that BL-FNSOM has fewer inactive neurons andcan obtain more effective map reflecting the distribution stateof input data than BL-SOM.

2008 International Joint Conference on Neural Networks (IJCNN 2008)

R EFERENCES[1] J. Vesanto and E. Alhoniemi, “Clustering of the Self-Organizing Map,”
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2008 International Joint Conference on Neural Networks (IJCNN 2008)

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