平成19年度電気関係学会四国支部連合大会

356

17-26

Neural Gas
Containing Two Kinds of Neurons
Keiko KANDA

Haruna MATSUSHITA(Tokushima University)

1. Introduction
Recently, the Neural Gas network[1] has attracted
attention as a image-recognition technology. In thelearning, Neural Gas uses one type of neurons. However, in the real world, an elephant, which has a slow
heart rate, and a mouse, which has a rapid heart rate,
have diﬀerent senses of time. In this study, we propose
a new Neural Gas algorithm; Neural Gas containing
two kinds of neurons. One converges rapidly, and theother converges slowly.

2. Algorithm of Proposed Method
We explain the algorithm of the proposed method.
The proposed method has two kinds of neurons; A1and A2 , whose features are diﬀerent. The set A1 contains m1 neurons, and the A2 contains m2 neurons.
Therefore, the network consists of a set A containing
M = m1 + m2 neurons; A = {c1 , c2 , · · · , cM }. The
neurons of the set A are classiﬁed into A1 and A2 atrandom. Each neuron ci has a d-dimensional weightvector wi = (wi1 , wi2 , · · · , wid ) (i = 1, 2, · · · , M ). The
initial values of the weight vectors are given at random.
The range of the elements of d-dimensional input data
xj = (xj1 , xj2 , · · · , xjd ) (j = 1, 2, · · · , N ) are assumed
to be from 0 to 1.

Yoshifumi NISHIO

where λi and λf are the initial and the ﬁnal valuesof λ(t), respectively. As a result of Eq. (5), λ of A1
decreases rapidly. On the other hand, the parameter
λ of A2 decreases linearly. In other words, the neuronsof A1 converge rapidly, and the neurons of A2 converge
slowly.
(TK-NG4) If t < T , return to (TK-NG1).

3. Computer Simulation
We consider 2-dimensional input data as shown inFig. 1(a). The input data has 1 cluster and noises.
There are 700 points in the cluster and 300 points fornoises. The simulation result of the conventional Neural Gas is shown in Fig. 1(b). We can see that there
are some neurons not only inside of the cluster but
also outside. The other side, the result of the proposed Neural Gas is shown in Fig. 1(c). We conﬁrm
that there are A2 neurons in the cluster and A1 neurons are outside of the cluster. In other words, proposed method learns by using diﬀerent kind of neurons
in the cluster and noises, while conventional method
uses only one kinds of neurons. We can see from thisﬁgure that the proposed Neural Gas holds the possibility to an application to the noise reduction.
10.9(TK-NG1) Euclid distances dji between an input
vector xj and all the weight vectors wi are calculated;

0.80.70.6

dji = kxj − wi k.

(1)

0.5

0.4

(TK-NG2) Each neuron ci is assigned a rank k =0, · · · , (M − 1) depending on their dji . The rank of
0 indicates the neuron closest to xi and the rank of(M − 1) indicates the neuron farthest from xi .
(TK-NG3) The weight vectors of the neurons are updated according towi (t + 1) = wi (t) + E(t)hλ (t)(xj − wi (t)),
where E(t) is the learning rate;( ) TtEi
E(t) = Ei,Ef(2)

(3)

0.2

0.1

0(4)where the parameter λ(t) controls the width of theneighborhood function;
( ) Ttλf,ci ∈ A1
 λiλiλ(t) =(5))(tt λi 1 −+ λf , ci ∈ A2 ,TT

0

0.2

0.4

0.6

0.8

1(a)
1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0

where t is the time step and T is the total number oftraining steps. Ei and Ef are the initial and the ﬁnal
values of E(t), respectively. hλ (t) is the neighborhood
ranking function described as;hλ (t) = exp(−k/λ(t)),

0.3

0.1

00.2

0.4

0.6

0.8

1000.2

0.4

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0.8

1(b)
(c)
Figure 1: Learning simulation for 2-dimensional input data.
(a) Input data. (b) Result of conventional Neural Gas. (c) Result of proposed Neural Gas.

4. Conclusions
We have proposed a new Neural Gas algorithm;
Neural Gas containing two kinds of neurons. We have
conﬁrmed that there was the possibility of application
to the noise reduction by using two kinds of neurons.
Reference[1] T. M. Martinetz and K. J. Schulten, “Neural-gas network
learns topologies. In T. Kohonen, K. Makisara, O. Simula, andJ. Kangas, editors”, Artiﬁcal Neural Networks, pp. 397-402,
1991.