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LETTER A Topology Preserving Neural Network for Nonstationary Distributions

LETTER

A Topology Preserving Neural Network for Nonstationary Distributions

Taira NAKAJIMA?,Hiroyuki TAKIZA W A?,Nonmembers,Hiroaki KOBAYASHI?,

and Tadao NAKAMURA?,Members

SUMMARY We propose a learning algorithm for self-organizing neural networks to form a topology preserving map from an input manifold whose topology may dynamically change. Experimental results show that the network using the proposed algorithm can rapidly adjust itself to represent the topology of nonstationary input distributions.

key words:competitive Hebbian learning rule,law of the jungle mechanism,neural networks,nonstationary probability distribu-tion,self-organizing map.

1.Introduction

A topology preserving map plays an important role in neural processing systems,and a wide variety of stud-ies from theory to application have been performed[1]. Usually,a neural network(NN)to form topology pre-serving maps consists of nodes i,i∈{1,...,N}with lateral connections,each of which determines the neigh-borhood relationship between two nodes.These nodes and their connections form a graph G to represent a topological structure.Each node has a synaptic weight vector w i∈ D to indicate a point of a D-dimensional input space.

The NN receives a series of input vectors x∈ D where the probability of their occurrence obeys some unknown probability distribution function(PDF). Then,the NN forms a map to project input vectors onto the nodes of the NN where similar input vectors are projected onto adjacent nodes,and vice versa.More precisely,the NN forms a topology preserving map from the manifold M? D,to the graph G,and vice versa. Here,M consists of regions where probability of input occurrence is non-zero.

Kohonen’s Self-organizing map(SOM)algorithm is the most popular learning algorithm to form a topol-ogy preserving map[5].However,in the case where the topology of the SOM network is di?erent from that of an input manifold,the SOM cannot generally form a topology preserving map because the SOM network keeps a?xed topology during learning.

Martinetz proposed the Competitive Hebbian learning rule(CH)[3]to form a perfectly topology pre-serving map.The CH assumes that the density of Manuscript received September14,1998.

?The authors are with the Department of Computer and Mathematical Sciences,Graduate School of Information Sci-ences,Tohoku University,Sendai-shi,980–8579Japan.weight vectors of an NN is in proportion to the proba-bility density of input occurrence over an input space. Thus,the NN is considered a vector quantizer(VQ)of an unknown PDF.For a given unknown PDF,a rep-resentation of the PDF using a VQ is obtained?rst, and then the CH is used to generate an appropriate topology.Therefore,the CH and the VQ are processed separately.For fast topology preservation learning,it would be e?ective to apply both the CH and the VQ concurrently to the formation of a topology preserv-ing map.Several works are proposed considering the combination of a neural VQ and the CH[4],[6],[7],and successfully make NNs form topology preserving maps during the approximations of PDFs.

However,all of the neural VQs used in topol-ogy preserving NNs are not suitable for approximating nonstationary PDFs because their VQs require time-decaying parameters that pre-de?ne the total number of learning cycles,or need periodic adaptations that have no relation to changes in PDFs.Hence,the lack of ability for approximating nonstationary PDFs pre-vents their NNs from forming topology preserving maps in nonstationary environments.

As nonstationary environments are more common than stationary environments in the real world,it is important for topology preserving NNs to form a map from nonstationary PDFs.Promising applications for the proposed algorithm would be real time feature ex-traction from video sequences,data analysis in nonsta-tionary environments,and so on.

Recently,we proposed a method that can improve the Kohonen learning(KL)[5],named the law of the jungle(LOJ).The LOJ allows neural VQs to approx-imate nonstationary PDFs[2].As the LOJ does not have topological connections among nodes,the NN us-ing the LOJ cannot form a topology preserving map. However,the integration of the LOJ and the CH may lead us to a new NN that can form topology preserving maps from nonstationary PDFs.

In this paper,we discuss the possibility of combin-ing the LOJ and the CH,and propose an algorithm for NNs that can form a topology preserving map from a manifold M whose topology may change.In Sect.2, we brie?y mention the foundations of the CH and the LOJ.In Sect.3,we?rst discuss a method to combine the LOJ and the CH,and then propose an algorithm

for NN that can form a topology preserving map from

a manifold M whose topology may change.Section4 gives the experimental results and discussion.We con-clude our work in Sect.5.

2.Foundations

In this section,we brie?y describe the foundations of the CH and the LOJ.

2.1The Competitive Hebbian Learning Rule

The CH allows a neural VQ to form a perfectly topology

preserving map from an input manifold M.The CH as-sumes that an NN consists of nodes i,i∈{1,...,N}

with weight vectors w i∈ D.It also assumes the weight vectors are distributed to approximate the input

manifold M,for any w i,w i∈M.The CH successively inserts topological connections between nodes by using the following way:

For each input x,connect two nodes which

have the nearest and the second nearest

weights measured by Euclidean distance.

This procedure is based on the idea that nodes with

weight vectors having neighboring masked Voronoi polygons are connected[3].Martinetz showed that the resulting graph G of the nodes with the lateral connec-tions forms“induced Delaunay triangulation”of the weight vectors w i,and hence the CH can construct perfectly topology preserving maps of arbitrarily struc-tured input manifolds[3].There are several methods to concurrently use the CH and the VQ to achieve a fast topology learning as mentioned in the previous section. In these cases,they have to use techniques for remov-ing obsolete lateral connections between nodes since the movement of the weight vectors in the VQ may make the connections invalid.

Though these methods to use the CH with the VQs

can form topology preserving maps during approxima-tions of input PDFs,all of the VQs are not suitable for approximating nonstationary PDFs,because these VQs require time-decaying parameters that pre-de?ne the total number of learning cycles,or need periodic adaptations that have no relation to changes in PDFs. Hence,the lack of ability for approximating nonstation-ary PDFs prevents their NNs from forming topology preserving maps in nonstationary environments.

2.2The Law of the Jungle Mechanism

The KL was proposed to approximate an unknown PDF[5].Within the KL,for an input vector,a node with the nearest weight vector from the input vector measured by Euclidean distance is selected as the win-ner node through a competition.The learning is ac-complished by moving the weight vector of the winner node toward the input vector.However,a node with the weight vector far from any input may never win, therefore never learn.From the above reason,the KL would not generally form the optimal set of weight vec-tors,where all the nodes should win the competition equally.

The LOJ is proposed to allow the KL to approxi-mate nonstationary PDFs[2].In the LOJ,the adapta-tion of the weight vectors is usually performed in the same way of the KL.To accelerate the convergence speed,the following process is employed in the LOJ. Results of the competitions are stored in the winner history table,which can hold up to M entries,and from the table the win probability of each node is dynami-cally estimated during learning.By using the estimated win probabilities,“strong”nodes are increased through creating new nodes near the nodes,and“weak”nodes are decreased through deleting them.A pair of the creation and the deletion is treated as an atomic op-eration.Therefore,the nodes which cannot win the competition are transferred directly from the regions where inputs seldom occur to the regions where inputs frequently occur.These direct“jumps”of the weak nodes to the strong nodes provides rapid convergence of the KL.Moreover,the LOJ requires neither time-decaying parameters nor special periodic adaptations, these properties are essential for dealing with nonsta-tionary PDFs[2].Like ordinary VQs,the LOJ has no topological relationships among nodes,therefore the NN using the LOJ cannot form a topology preserving map.However,concurrent use of the LOJ and the CH may lead us to a new NN that can form topology pre-serving maps from nonstationary PDFs,which cannot be achieved by the ordinary VQs with the CH.

3.Topology Preserving Network in Nonsta-

tionary Environments

In this section,we?rst discuss possibility of integrating the LOJ and the CH,and then describe a learning al-gorithm for an NN to form a topology preserving map between a network G and a manifold M whose topol-ogy may change during learning.We call the algorithm the LOJ combined with the CH(LOJ/CH).

3.1Integration of the LOJ and the CH

To consider the combination of the LOJ and the CH,we must discuss problems from two kinds of view points: 1.Problems caused by the CH in the process of the

LOJ to approximate a nonstationary PDF.

2.Problems caused by the LOJ in the process of the

CH to form topological connections between nodes. The CH only makes connections between nodes,which does not disturb the movement of weight vectors.More-over,the CH requires neither time-decaying parame-

LETTER

1133

ters nor special periodic adaptations,which prevent the NNs from dealing with nonstationary PDFs.As a re-sult,the CH has no problem in the integration with the LOJ.

The problems of the LOJ is the movement of weight vectors,because only the movement of weight vectors makes the topological connections between nodes invalid.Therefore,we should discuss a method to eliminate obsolete connections caused by the move-ment of weight vectors.Within the LOJ,weight vec-tors could move in two di?erent ways.One is due to sequential adaptation toward inputs,which is also re-quired in most VQs.However,obsolete connections caused by the sequential adaptation can be eliminated with the edge aging scheme[4].The other is due to the direct jumps of the weight vectors of weak nodes toward the weight vectors of strong nodes,which are peculiar to the LOJ.These direct jumps usually move the weights drastically on the manifold.As a result, the jumps make topological relationships among nodes invalid.Therefore,the connections between a node and its neighbors should be removed when the node jumps. This removal may make topology learning slow down, but fortunately,the jumps would be performed only in the initial adaptation stage where win probabilities of nodes are quite di?erent from each other.Therefore, this slowdown caused by the removal could be ignored.

3.2Learning Procedure of the LOJ/CH

The network of the LOJ/CH consists of nodes i,i∈{1,...,N},each of which has a synaptic weight vector w i∈ D to indicate a point of a D-dimensional input

space.Any two of the nodes can form a lateral connec-tion which determines the neighborhood relationship between the two nodes.These nodes with connections form a graph G to represent a topological structure.All the connections have ages to be used to delete invalid connections whose ages exceed some certain value.The network has the winner history table which can store up to M indices of the winner nodes.

Table1shows the learning procedure of the LOJ/CH.In the?rst step,a random weight is given to each node.In the second step,a new input x∈ D gen-erated according to some unknown PDF is presented to the network.In the third step,the winner node j that is the nearest node from the input measured by Eu-clidean distance and second-winner node k that is the second-nearest node from the input are selected.

In the forth step,the age of each connection ema-nating from the winner node j is incremented.In the ?fth step,if nodes j and k have no connection between them,a new connection is created.The age of the con-nection between j and k is initially set to zero.In the sixth step,connections whose ages are larger than a max are removed.

Steps seventh through ninth are completely same

Table1Learning procedure of the LOJ/CH.

Step1.Initialize weight of each node.

Step2.Present new input.

Step3.Select winner node j and second-winner node k. Step4.Increment ages of connections.

Step5.Adjust connection for node j and k.

Step6.Delete old connections.

Step7.Adjust winner history table.

Step8.Deduce win probability.

Step9.Decide whether creation and deletion is performed.

If creation and deletion is needed,go to step11. Step10.Adapt weights of the winner node and its

neighbors.Go to step2.

Step11.Create new node for the winner node,and delete the weakest node.Adjust winner history table.

Go to step2.

as the original LOJ[2].

The tenth step is performed only if creation and deletion is not performed.In this step,the weight vec-tors of the winner node and its neighbors are adjusted using Eq.(1).

w j=w j+αw(x?w j),

w n

i

=w n

i

+αn(x?w n

i

),

i∈{(neighbors of j)},(1)

where w j,and w n

i

are the weight vectors of the winner node and its neighbors,respectively.The learning rates for the weight adjustment of the winner node and its neighbors areαw andαn,respectively.This is a general process in sequential weight adaptation[6],[7].Then the learning goes back to step2.

In the eleventh step,creation and deletion of nodes is performed.Deletion of a node is performed by delet-ing the weakest node l and its connections.In addition, the entries of node l are removed from the winner his-tory table.Creation of a node is described as follows. To clarify the explanation,let node j be the winner node before the creation,and node j be the winner node after the creation.Node creation begins with cre-ating a new node j new whose weight vector points the position calculated by Eq.(2).

w j

new

=w j+αw(x?w j).(2) This equation means that a new node j new is created in the same position of the winner node j and then the new node moves toward the input instead of the winner node j .The key point of the equation is that the jump and the adaptation of the node are simultaneously performed.To avoid pointing an identical position from plural nodes,the winner node j does not move in this process.Next process is to make proper connections between:

?node j new and its neighbors,

?node j and its neighbors,and

?node j new and j .

The process of the CH is to connect nodes whose

1134

IEICE TRANS.INF.&SYST.,VOL.E82–D,NO.7JULY

1999

(a)Simple

(b)Clusters

(c)Complex

Fig.1Input manifold.

weight vectors having neighboring masked Voronoi polygons [3],that is,to connect two adjacent nodes on the connected input manifold M .Thus,we obey this policy to connect these two adjacent nodes on M to make or change the connections.We assume that two nodes j new and j are on the connected manifold M .First,a new connection between nodes j new and j with age “zero”is made.Next,if the following condition is satis?ed,each connection emanating from node j is changed to the connection emanating from node j new with keeping the same age.

||w j new ?w n i ||<||w j ?w n i ||,

(3)

where the norm is Euclidean distance and nodes n i are neighbors of j .The last process of the creation is to replace the entries of node j in the winner history table with those of nodes j new and j alternately.This is because nodes j new and j share the space that was previously occupied by node j .

The following learning is accomplished by iterating steps 2through 11while inputs are presented.4.

Experimental Results and Discussions

In this section,we perform some experiments to exam-ine the performance of the LOJ/CH.Figure 1shows the input manifold M ? 3used in the experiments.

The input sequence consists of three phases,i.e.,the manifold changes twice at T =10001and T =20001.Here,T is the number of input vectors pre-sented (learning steps).In the ?rst phase (from T =1to T =10000)named “simple,”the manifold is a sim-ple 2-dimensional square,which is used to con?rm the ability of NNs to distribute weights and form a map.In the second phase (from T =10001to T =20000)named “clusters,”the manifold consists of several 3-dimensional balls for examination of the clustering abil-ity of NNs.In the last phase (from T =20001to T =30000)named “complex,”the manifold consists of a 2-dimensional torus and a 1-dimensional ring for

examination of the ability of NNs to represent a “com-plex”topology of the manifold M .For comparison,the SOM are examined by using the same input manifold.Since the SOM is not designed to be used in nonsta-tionary environment,we cannot directly apply a SOM network to the experiments.For the SOM network,we therefore divide the input sequence into three station-ary phases,and individually measure each performance as three sub-experiments (each learning length =10000steps).

The SOM network has a 2-dimensional (?xed)topology,and the number of the nodes Nis 100(10×10).The initial learning rate for the winner node at T =1is 0.1and linearly decays to 0at T =10000.The neighboring function is the “bubble”function [1]and the radius to determine the neighbor is 10at T =1and linearly decays to 1at T =10000.In the net-work using LOJ/CH,following parameters are used:N =100,αw =0.1,αn =0.003,a max =50,M =1800,Cr =3.5,De =0.05.Within the learning process of the LOJ/CH,creation and deletion of the nodes are inhibited in the initial leaning stage of T =1through 1000,because the insu?cient number of entries in the winner history table in the initial learning stage make the network behavior unstable.

We compare the LOJ/CH and the SOM in terms of approximation of PDFs and formation of the topol-ogy preserving map.We use the mean square error (MSE)for metric of approximating PDFs.In general,an appropriate metric of topology preservation consists of two parts.One is the degree of topology preserva-tion of a map from a network G to an manifold M ,and the other is the degree of preservation from M to G .Although we need a metric dealing with a net-work G having a variable topology,there only exists a metric to calculate the degree of topology preserva-tion from G to M [8].Therefore,we use an imperfect metric TPG=ΦM G (?1)that is a part of a topological function proposed in [8].The TPG calculates the num-ber of connections which violate topology preservation

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(a)Simple

(b)Clusters

(c)Complex

Fig.2Node distributions.

Table 2MSE of LOJ/CH and SOM in each phase.

NN

Simple Clusters Complex LOJ/CH 0.200.360.51SOM

0.27

0.57

0.83

Table 3TPG of LOJ/CH and SOM in each phase.

NN

Simple Clusters Complex LOJ/CH 2.6 5.3 3.0SOM

0.0

87

87

in the map from G to M .

Tables 2and 3show the MSE and the TPG at the end of each phase,respectively.The results are mean values during ten trials.Figure 2shows a typi-cal nodes distribution with connections of LOJ/CH at the end of each phase.In all phases,the MSEs of the LOJ/CH are superior to those of the SOM including the simple phase,even though the simple phase has the same topology of the SOM network.Remarkable dif-ference between them in the clusters phase shows that the LOJ/CH would be favorable in clustering.Fig-ure 2(b)also supports this observation:all the clusters are properly separated.Since the topologies of the in-put manifold in the simple phase and the SOM network are the same,the TPG of the SOM is perfect.However,the TPGs of the SOM in the other phases are remark-ably poor.On the other hand,even though movement of weight vectors prevent the LOJ/CH from forming perfectly topology preserving maps,the TPGs of the LOJ/CH are quite smaller and more stable than those of the SOM in all phases.Especially,the TPG in the complex phase is almost the same as that in the simple phase.Figure 2(c)also shows that the LOJ/CH net-work can represent the complex topology in the com-plex phase.As a result the ability of the LOJ/CH to represent topology of input manifold would be useful for approximation problems with complex manifolds even if the topology changes.

5.Conclusions

We have proposed a learning algorithm named LOJ/CH for an NN to form a topology preserving map from an input manifold whose topology may change.The LOJ/CH is a combination of one of the VQ algorithms “LOJ,”which can approximate nonstationary PDFs,and the topology generating algorithm “CH,”which can form a perfectly topology preserving map from arbi-trary input manifold.Experimental results show that the LOJ/CH can rapidly form a topology preserving map even if the topology of input manifold changes.The LOJ/CH also has the ability to clusterize the in-put space in nonstationary environments.

We are now planning to apply the LOJ/CH to the feature extraction from medical image sequences.

References

[1]T.Kohonen,“Self-Organizing Maps,”Springer-Verlag,

Berlin Heidelberg,1995.

[2]T.Nakajima,H.Takizawa,H.Kobayashi,and T.Nakamura,

“Kohonen learning with a mechanism,the law of the jungle,capable of dealing with nonstationary probability distribu-tion functions,”IEICE Trans.Inf.&Syst.,vol.6,pp.584–591,1998.

[3]T.M.Martinetz,“Competitive Hebbian learning rule forms

perfectly topology preserving maps,”Proc.Int.Conf.on Ar-ti?cial Neural Networks,pp.427–434,1993.

[4]T.M.Martinetz and K.J.Schulten,“Topology representing

networks,”Neural Networks,vol.7,no.3,pp.507–522,1994.[5]T.Kohonen,“Self-Organization and Associative Memory,”

Springer-Verlag,Berlin Heidelberg,1989.

[6] B.Fritzke,“A growing neural gas network learns topologies,”

Advances in Neural Info.Proc.Systems,vol.7,pp.625–632,1995.

[7]J.Bruske and G.Sommer,“Dynamic Cell Structure Learns

Perfectly Topology Preserving Map,”Neural Computation,vol.7,pp.845–865,1995.

[8]T.villmann,R.Der,M.Herrmann,and T.Martinetz,

“Topology preservation in self-organizing feature maps:Ex-act de?nition and measurement,”IEEE Trans.Neural Net-works,vol.8,no.2,pp.256–266,1997.

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