Lazy Self-Organizing Map and its Behaviors
Taku Haraguchi, Student member, IEEE, Haruna Matsushita, Student member, IEEE,and Yoshifumi Nishio, Member, IEEE
Abstract— The Self-Organizing Map (SOM) is a famousalgorithm for the unsupervised learning and visualization introduced by Teuvo Kohonen. This study proposes the Lazy SelfOrganizing Map (LSOM) algorithm which reﬂects the world ofworker ants. In LSOM, three kinds of neurons exist: worker
neurons, lazy neurons and indecisive neurons. We apply LSOM
to various input data set and conﬁrm that LSOM can obtain amore effective map reﬂecting the distribution state of the input
data than the conventional SOM.
Worker ants

I. I NTRODUCTIONLUSTERING is one of typical analysis techniques andis studied for many applications, such as statement,
pattern recognition, data mining. In recent years, the SelfOrganizing Map (SOM) [1] has attracted attention for the
study on clustering [2]. SOM is an unsupervised neuralnetwork introduced by Kohonen in 1982 and is a simpliﬁed
model of the self-organization process of the brain. SOM
can classify input data according to similarities, which are
obtained by the distance between neurons, and is applied towide ﬁelds of data classiﬁcations. In the learning algorithmof SOM, a winner neuron and its neighboring neurons are
updated. A learning rate, which denotes the updating degreeof neurons, decreases with time, and the neuron nearer tothe winner is signiﬁcantly updated. This means that all theneurons can be updated, in so far as it is near to the input
data or the winner neuron. In other words, all the neurons of
the conventional SOM are good worker neurons.
Meanwhile, it is believed that God does not create an unnecessary thing even if it seems to useless. There is a reportthat 20 percent of worker ants are “lazy” (as Fig. 1) [3].
These ants keep still or stay around their nests. However, theresearchers think that the lazy ants have certain role. Because,
in another experiment, the ants group, which includes thelazy ants at food collections, can collect more foods than
the group which includes only the worker ants. The worker
ants can collect food efﬁciently, however, it may be hard toﬁnd new foods just because they consider the efﬁciency. In
addition, it is thought that chances of ﬁnding foods increase
because the lazy ants maunder without working. In otherwords, we can obtain better results in the case that uselessthings exist than only excellent things.
In this study, we propose a new type of SOM algorithm,
which is called Lazy SOM (LSOM) algorithm. The importantfeature of LSOM is that three kinds of neurons exist; worker

C

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

Lazy ants

Fig. 1.

neurons, lazy neurons, which do not work and indecisive
neurons which are the neighborhoods of the lazy neurons.
The learning rate of the lazy neurons is smaller than the onesof the worker neurons. The learning rate of the indecisive
neurons become small due to the lazy neurons. We can saythat LSOM carries out learning reﬂecting the world of worker
ants.
We explain the learning algorithm of LSOM in detail in
Section III. In Section IV, we apply LSOM to 2-dimensionaland 3-dimensional input data, which have some clustering
problems. Furthermore, we explain the learning behaviorsof LSOM in detail. Learning performances are evaluatedboth visually and quantitatively using two measurements
and are compared with the conventional SOM. We conﬁrm
that LSOM can obtain a more effective map reﬂecting the
distribution state of the input data than the conventional
SOM.
II. S ELF -O RGANIZING M AP (SOM)
SOM has a two-layer structure of the input layer and thecompetitive layer. In the input layer, there are d-dimensional
input vectors xj = (xj1 , xj2 , · · · , xjd ) (j = 1, 2, · · · , N ).
In the competitive layer, M neurons are arranged on the 2dimensional grid. Each neuron has a weight vectors w i =(wi1 , wi2 , · · · , wid ) (i = 1, 2, · · · , M ) with the same dimension as the input vector. The range of the input vector isassumed to be between 0 and 1. The initial values of all theweight vectors are given between 0 and 1 at random.

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World of worker ants. Some worker ants are “lazy”.

Start

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Indecisive neuron

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Worker neuronᵒLazy neuron

Input

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Find winnerᵒc ∈ S lazy
Yes

Indecisive neurons increase
with time !!

No

Find f andChange c for f

Fig. 2. LSOM contains three kinds of neurons: worker neuron, lazy neuron
and indecisive neuron. Each number denotes neighborhood distance between
lazy neuron and each neuron.

No

D ≥1
Yes

(SOM1) We input an input vector xj to all the neurons atsame time in parallel.
(SOM2) We ﬁnd a winner neuron by calculating the distances between the input vector xj and the weight vector
wi of neuron i. The winner neuron c is the neuron with theweight vector nearest to the input vector xj ;c = arg min{wi − xj },iFind N lazyj = j +1

Choose αUpdate weight vectors
No

(1)

Yes
No

where  ·  is the distance measure, in this study, we use
Euclidean distance.
(SOM3) The weight vector of all the neurons are updated as
w i (t + 1) = wi (t) + hc,i (t)(xj − w i (t)),

t <T
Yes
End

(2)

where t is the learning step. hc,i (t) is called the neighborhood
function and is described asri − r c 2
hc,i (t) = α(t) exp −,(3)
2σ 2 (t)where r i and r c are the vectorial locations on the display
grid, α(t) is called the learning rate, and σ(t) corresponds
to the widths of the neighborhood function. Both α(t) andσ(t) decrease monotonically with time, in this study, we use
the following;




t
t
, σ(t) = σ(0) 1 −,(4)α(t) = α(0) 1 −TTwhere T is the maximum number of the learning.
(SOM4) The steps from (SOM1) to (SOM3) are repeated for
all the input data.
III. L AZY S ELF -O RGANIZING M AP (LSOM)In this study, we propose Lazy SOM (LSOM). The important feature of LSOM is that three kinds of neurons exist(as Fig. 2): worker neurons, lazy neurons, which do notwork, and indecisive neurons which are neighborhoods ofthe lazy neuron. The updating degree of the lazy neurons is

Fig. 3.

Flowchart of LSOM learning.

smaller than the ones of the worker neurons. Furthermore,
the updating degrees of the lazy neurons’ neighbors become
small due to the lazy neuron, hence the name is “indecisive”.
The lazy neuron becomes the worker neuron whenever it
becomes the winner, and another neuron becomes lazy. The
number of the indecisive neuron increases with time.
A. Learning Algorithm
We explain the learning algorithm of LSOM in detail. A
ﬂowchart of the learning algorithm is shown in Fig. 3. In
LSOM, M neurons are arranged as a regular 2-dimensional
grid. p neurons are classiﬁed into a set of the lazy neurons
Slazy at random.
(LSOM1) An input data xj is inputted to all the neurons at
the same time in parallel.
(LSOM2) We ﬁnd the winner neuron c according to Eq. (1).
If c ∈ Slazy , we perform (LSOM3). If not, we perform(LSOM4).
(LSOM3) The lazy neuron, which is the winner c, is ex-

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This neuron is the furthest to input,
namely it becomes Lazy neuron !!
Lazy neuron

Lazy neuron

Worker neurons

Worker neuron

Lazy neuron becomes
worker neuron !!

Lazy neuron
becomes winner !!

xj

Fig. 4.

xj

Lazy neuron becomes worker neuron whenever it becomes winner, and another neuron becomes lazy.

cluded from the set of the lazy neuron Slazy , and a neuron
f , which is farthest from the input data xj and is not in
Slazy , is selected to become a member of Slazy ;f = arg max{wi − xj },ii ∈ Slazy .

(5)

In other words, the lazy neuron becomes worker if it becomes
the winner c, and another neuron f becomes lazy shown as
Fig. 4.
(LSOM4) We ﬁnd the indecisive neurons. A set of theindecisive neurons Nlazy are the neighborhoods of each lazyneuron l in Slazy .
Nlazy = {i | ri − rl 2 ≤ D(t),i = c, i ∈ Slazy , l ∈ Slazy },(6)

where ||r i − rl || is the neighborhood distance between map
nodes i and l on the map grid, and D(t) corresponds tothe neighborhood size, namely the quantity of the indecisive
neurons. D(t) increases with time according to the following
equation;


t
,
(7)
D(t) = Dmax
T
where [ ] denotes the Gauss’ notation and Dmax is a ﬁxed
parameter deciding the max value of D(t).
(LSOM5) The weight vectors of all the neurons are updatedas(8)
w i (t + 1) = wi (t) + hLc,i (t)(xj − w i (t)),
where the function hLc,i (t) is the neighborhood function of
LSOM and is described asri − r c 2,
(9)hL c,i (t) = α exp −2σ 2 (t)where α is the learning rate of LSOM and is decided by eachneuron’s character;⎧⎨ αlazy , if i = l, l ∈ Slazy
αN ,if i ∈ Nlazy
(10)
α=
⎩
αw ,
otherwise,
where αw is the learning rate of the worker neurons, αlazy is
the learning rate of the lazy neurons, and αN is the learning

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rate of the indecisive neurons, namely αlazy ≤ αN ≤ αw .
(LSOM6) The steps from (LSOM1) to (LSOM5) are repeated for all the input data.
IV. E XPERIMENTAL R ESULTS
A. For Target data
We consider a 2-dimensional input data: Target data set
shown in Fig. 5(a). This data contains 770 points which has
a clustering problem of outliers [4]. Both the conventional
SOM and the proposed LSOM have 225 neurons (15 × 15)
each. p=20 neurons, namely 20% of the neurons, are classiﬁed into a set of the lazy neurons. We repeat the learning 20
times for all input data, namely T = 15400. The parameters
for the learning are chosen as follows;(For SOM)

α(0) = 0.3, σ(0) = 5.0,

(For LSOM)
αw = 0.3, αlazy = 0.03, αN = 0.15
σ(0) = 5.0, Dmax = 5.
where we use the same learning rate: α(0) and αw , and σ(0)for the comparison and the conﬁrmation of the lazy neuron
effect.
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Fig. 5. Learning simulation for Target data. (a) Input data. (b) Learning
result of the conventional SOM.

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Fig. 6. Learning process of proposed LSOM. Red points, blue points and white points denote worker neurons, lazy neurons and indecisive neurons,
respectively. (a) Initial state (t = 0). (b) t = 500. (c) t = 4000. (d) t = 9000. (e) t = 11000. (f) t = 12000. (g) t = 13000. (h) Learning results (t = 15400).

The simulation results of the conventional SOM is shownin Fig. 5(b). We can see that the conventional SOM does not
self-organize up to all the outliers input data. Figures 6(a)-(h)show the learning process of LSOM, and the ﬁnal result isshown in Fig. 6(h). We can see that LSOM can self-organize
up to all the corner data. This is because LSOM has threekinds of neurons reﬂecting the world of worker ants.
1) Behavior of LSOM: Let us consider the learning process and behaviors of LSOM in more detail. In the earlystage of learning as in Fig. 6(b), many worker neurons tend
to gather at concentrated input data, namely the cluster area.
Meanwhile, the lazy neurons tend to move to the outside ofcluster because they are not signiﬁcantly updated. However,as learning progresses, the lazy neurons self-organize othercluster as in Fig. 6(c) because they are in different area fromthe worker neurons. That is to say the lazy neurons candiscover new input data since they maunder. In the middle
stage of learning as Fig. 6(d), the indecisive neurons, whoselearning rate are larger than the lazy neurons but smaller
than the worker neurons, increase. The lazy neurons andthe indecisive neurons self-organize outside the cluster. As
more learning progresses, the lazy neurons self-organize theoutliers as in Fig. 6(e). In the last stage of learning as inFigs. 6(f)-(h), the number of indecisive neurons increase
further, and the map converges. From these ﬁgures, we cansay that LSOM can self-organize in every corner of input
data than the conventional SOM.
2) Comparison between SOM and LSOM: Furthermore,in order to compare the learning performance of LSOM withthe conventional SOM numerically, we use the following

well-used two measurements.
Quantization Error Qe [1]: This measures the average
distance between each input vector and its winner;
N

Qe =

1
xj − w̄j ,
N j=1

(11)

where w̄j is the weight vector of the corresponding winnerof the input vector xj . Therefore, the small value Qe is moredesirable.
Neuron Utilization U [5]: This measures the percentage ofneurons that are the winner of one or more input vector in
the map;
nm
1
ui ,
(12)
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 two measures are shown in Table I. The
quantization error Qe of LSOM is smaller than the conventional SOM, and by using LSOM, the quantization error Qe
has improved by 45.3545% compared to that of SOM. In
addition, the neuron utilization U of LSOM is larger thanthe conventional SOM, and by using LSOM, the neuron
utilization U has improved by 11.6959%. This result meansthat the result of LSOM have fewer inactive neurons than theconventional SOM, and LSOM can obtain more exact map
reﬂecting the distribution state of input data.

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Learning simulation for Atom data. (a) Input Data. (b) Learning simulation of conventional SOM. (c) Learning simulation of LSOM.

TABLE ITABLE IIQ UANTIZATION ERROR Qe AND N EURON UTILIZATION U FOR TARGETQ UANTIZATION ERROR Qe AND N EURON UTILIZATION U FOR H EPTADATA .

DATA .

Conventional SOM
LSOM
Improvement rate [%]Qe
0.01470.008045.3545U
0.76000.848911.6959

B. For Hepta data
Next, we considered a 3-dimensional input data called
Hepta data set shown in Fig. 7(a), which has a clustering
problem of different densities in clusters. The total numberof the input data N is 212, and the input data has sevenclusters.
We repeated the learning 70 times for all the input data,
namely T = 14840. The learning conditions are the same as
those used in Subsection IV-A.
Figures 7(b) and (c) show the learning results of theconventional SOM and LSOM, respectively. We can seethat LSOM can self-organize up to edge data than theconventional SOM.

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Conventional SOM
LSOM
Improvement rate [%]Qe
0.02790.013850.4575

U
0.53330.5600
5.0

The calculated two measures are shown in Table II. The
quantization error Qe of LSOM is smaller and the neuron
utilization U of LSOM is larger than the conventional SOM.
Because the inactive neurons have been reduced by 5.0%,
more neurons are attracted to clusters and the quantization
error Qe has been decreased by 50.4575%. It means that
LSOM can extract the feature of the input data more effectively than the conventional SOM.
C. For Atom data
Furthermore, we considered a 3-dimensional input data
called Atom data set shown in Fig. 8(a). This data set hasclustering problems of linear not separable, different densities

2008 International Joint Conference on Neural Networks (IJCNN 2008)

TABLE III
Q UANTIZATION ERROR Qe AND N EURON UTILIZATION U FOR ATOMDATA .

Conventional SOM
LSOM
Improvement rate [%]Qe
0.04930.031436.2946

U
0.74670.8267
10.7143and variances. The total number of the input data N is 800,
and the input data has two clusters.
We repeated the learning 20 times for all input data,
namely T = 16000. The learning conditions are the sameas those used in Subsection IV-A.
The learning results of the conventional SOM and LSOM
are shown in Figs. 8(b) and (c). We can see that LSOM canself-organize edge data more effective than the conventional
SOM.
Table. III shows the calculated two measures. The quantization error Qe of LSOM is smaller than the conventional
SOM, and the quantization error Qe has improved 36.2946%
from using the conventional SOM. In addition, the neuron
utilization U of LSOM is larger than the conventional SOM,
and by using LSOM, the neuron utilization U has improvedby 10.7143%. From these results, we can say that LSOM,
which includes the lazy neurons, can self-organize more
effectively than the conventional SOM which contains only
worker neurons.
V. C ONCLUSIONSIn this study, we have proposed the Lazy Self-Organizing
Map (LSOM), which reﬂects the world of worker ants.
LSOM contains three kinds of neurons: worker neurons, lazyneurons and indecisive neurons which are neighborhoodsof the lazy neurons. We have investigated the behaviorsof LSOM by applying it to some input data set. We have
conﬁrmed that LSOM can self-organize in every corner ofthe input data and can obtain more exact map reﬂecting the
distribution state of input data than the conventional SOM.
From these results, we can say that LSOM, which includes
the lazy neurons, can self-organize more effectively than theconventional SOM which contains only worker neurons.
R EFERENCES[1] T. Kohonen, Self-Organizing Maps, Berlin, Springer, vol. 30, 1995.
[2] J. Vesanto and E. Alhoniemi, “Clustering of the Self-Organizing Map,”
IEEE Trans. Neural Networks, vol. 11, no. 3, pp. 586–600, 2002.
[3] H. Hasegawa, “Optimization of GROUP Behavior,” Japan EthologicalSociety Newsletter, no. 43, pp. 22–23, 2004.
[4] A. Ultsch, “Clustering with SOM: U*C”, Proc. Workshop on SelfOrganizing Maps, pp.75-82, 2005.
[5] Y. Cheung and L. Law, “Rival-Model Penalized Self-Organizing Map,”
IEEE Trans. Neural Networks, vol. 18, no. 1, pp. 289–295, 2007.

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