The Neo-Fuzzy Autoencoder for Adaptive Deep Neural Systems and its Learning

Abstract

In this paper the autoencoder based on the generalized neo-fuzzy neurons is proposed. Also its fast learning algorithm was proposed. Such system can be used as part of deep learning neural networks. The proposed neo-fuzzy autoencoder is characterized by high learning speed and less number of tuned parameters in comparison with well-known approaches. The efficiency of proposed approach has been examined based on different benchmarks and real data sets.

I. INTRODUCTION

The task of compressing information that should be further processed, is one of main problem, which is solved in Data Mining. For solving this problem, a lot of approaches [ 1-5 ] are proposed, at that more of these methods comprehend the information processing in batch mode, when the fixed-sized data set is processed many times.

It is very important in order to the process of compression, the loss of information is minimal. Nowadays the approaches based on Deep neural networks (DNNs) [ 6-9 ] are widely used for solving the many tasks, which is connected with analysis of Big Data. As it can be seen from many researches the DNNs provide significantly better results than the conventional shallow neural networks.

The inherent part of DNN is, so-called autoencoder, which implements the compression of the input data and forms the input layers of the neural network.

As such autoencoders the multilayer associative “bottleneck” perceptrons or restricted Boltzmann machines, in which nodes are the elementary Rosenblatt perceptrons with the sigmoidal activation functions, are often used.

Unfortunately, the learning process of such autoencoders demands a large time spending and cannot be implemented in online mode.

In the connection with the intensive development of Data Mining, Data Stream Mining, Web Mining over recent years the development of high speed information compression systems is an important problem. Such systems have to process data in sequential mode (perhaps in online mode) as the real information processing systems need.

II. THE ARCHITECTURE OF NEO-FUZZY

AUTOENCODER

Fig. 1 shows the architecture of the proposed autoencoder, which is autoassociative “bottle-neck” modification of Kolmogorov’s neuro-fuzzy network [ 10-14 ] that implements the multiresolution approach and is the universal approximator according to the theorem of Kolmogorov-Arnold and YamNguen-Kreinovich.

It should be notice in [ 15, 16 ] the architecture, which nodes are the neo-fuzzy neurons (NFNs) [ 17 ], is considered. In spite of the simplicity of learning algorithm for synaptic weights, such system is abundant in the sense of the membership functions number.

Using the generalized neo-fuzzy neurons [ 18 ] instead of conventional NFN allows significantly to reduce the membership functions number and to introduce stacked NN [ 19 ]. Such stacked NN allows to simplify the architecture of autoencoder and in this way to speed up the learning process.

Therefore, autoencoder consists of two sequentially connected layers, which are implemented with the generalized neo-fuzzy neurons GNFN[ 1 ] and GNFN[ 2 ].

The sequence of input signals, which have to compress x(k ) =(k ( x1 (k ), x2 ),, xn (k ))T ∈ Rn , ( k = 1, 2, is a number of the observation or a current instant of time), is fed to GNFN[ 1 ].

GNFN[ 1 ] consists of the n multidimensional nonlinear synapses MNSi[ 1 ] , i = 1, 2,, n , where each of them has a one input, m outputs, h membership functions µl[i1] (xi (k )), l = 1, 2,, h and mh tuned synaptic weights w[j1li] , j = 1, 2,, m . signals

The output of GNFN[ 1 ] is the compressed vector of the y(k ) =),, ( y1 (k y j (k ),, ym (k )) ∈ Rm , m < n , T which simultaneously is output of the autoencoder. The signal y(k ) is fed to the inputs of GNFN[ 2 ], which contains m inputs, m

multidimensional nonlinear synapses MNS[j2] , where each of them has one input, n outputs, h membership functions µl[j2] ( y j (k )), l = 1, 2,, h and nh synaptic weights

Thus, considered autoencoder contains 2nmh tuned synaptic weights and (n + m)h membership functions that is significantly fewer than in the architecture in [ 20 ].

In the outputs of GNFN[ 2 ] the recovered signal xˆ(k ) = ( xˆ1 (k ),, xˆi (k ),, xˆn (k ))T is formed. In such manner the autoencoder is the autoassociative hybrid neo-fuzzy system of computational intelligence. Fig 1. The architecture of the proposed neo-fuzzy autoencoder.

The proposed system is implemented a nonlinear mapping in the form j=1 l =1

 i=1 l =1 xˆi (k ) = ∑ m h ∑ wi[l2j]µ [ 2 ]  n h  lj  ∑ ∑ w[j1li]µl[i1] (xi (k ))  ,  or in the matrix form xˆ(k ) = W [ 2 ]µ [ 2 ] (W [ 1 ]µ [ 1 ] (x(k ))) , where W [ 1 ] =  w2[ 11 ]1 W [ 2 ] =  w2[ 21 ]1   w1[ 11 ]1     wm[ 11 ]1   w1[121]     wn[211] w1[ 12 ]1 w2[ 12 ]1

 w[ 1 ]

m21 w1[221] w2[ 22 ]1

 wn[ 22 ]1         w1[1h]n  w2[1h]n  w2[1h]n  w[ 1 ] 

mhn  w1[h2m]  w2[2h]m  w2[2h]m  wn[2h]m  µ [ 1 ] (x(k )) = (µ1[ 11 ] (x1 (k )),µ 2[ 11 ] (x1 (k )),,µ h[ 11 ] (x1 (k )), ІІІ. THE LEARNING ALGORITHM FOR SYNAPTIC

WEIGHTS OF NEO-FUZZY AUTOENCODER

For the tuning the synaptic weights of GNFN[ 2 ] we can use the gradient procedure of minimizing the quadratic criterion in the form wi[l2j] (k ) =wi[l2j] (k −1) −η [ 2 ] (k ) =wi[l2j] (k −1) +η [ 2 ] (k )ei (k )µl[j2] ( y j (k )) ∂ei2 (k ) ∂wi[l2j] = where η [ 2 ] (k ) is learning rate parameter of the output layer, which is chosen accordingly to the condition in [ 20, 21 ] η [ 2 ] (k ) =(r[ 2 ] (k ))−1; r[ 2 ] (k ) =α r[ 2 ] (k −1) + µ [ 2 ] ( y(k )) 2 where 0 ≤ α ≤ 1 is forgetting factor.

For tuning the synaptic weights GNFN[ 1 ] the optimized backpropagation error procedure, which for uniformly distributed in the line of X-axis the triangular membership functions with centers xl[i1] yl[j2] can be write in the form w[j1li] (k ) =w[j1li] (k −1) +η [ 1 ] (k )ei (k )µl[i1] (xi (k ))wi[j2] (k ) where η [ 1 ] (k ) =(r[ 1 ] (k ))−1; r[ 1 ] (k ) =α r[ 1 ] (k −1) + µ [ 1 ] (x(k )) 2 ,  ( yl[,2j] − yl[−21], j )−1, if y j (k ) ∈[ yl[−21], j , yl[,2j] ],  h   wi[j2] (k ) =∑ wi[l2j] (k )  ( yl[,2j] − yl[+21], j )−1, if y j (k ) ∈[ yl[,2j] , yl[+21], j ],  .

l=1  0 otherwise  The proposed learning algorithm for synaptic weights of autoencoder is characterized by high speed and adjugate following and filtering properties.

IV. EXPERIMENTS

For effectiveness verification of the proposed neo-fuzzy autoencoder, the data sets were taken from UCI Repository [ 22 ]: Iris, Wine, Hayes-roth. Data set “Iris” contains 150 observations (Number of Attributes: 4) of 3 classes, Data set “Wine” contains 178 observations (Number of Attributes: 13) of 3 classes, data set “Hayes-roth” contains 160 observations (Number of Attributes: 5) of 3 classes.

It is seen from Fig.2 data, which are compressed using neofuzzy autoencoder, are more compact clusters than data, which are compressed based on the autoassociative multilayer neural network “Bottle Neck”.

a) b)

Fig. 2. Data set Hayes-roth after compression based on the autoassociative multilayer neural network “Bottle Neck” (а) and neo-fuzzy autoencoder (b)

The results, which were obtained using the proposed neofuzzy autoencoder, were compared with the results of autoassociative multilayer neural network “Bottle Neck” (Table I). The dimension of compression data was 2 components. The simulation was performed 20 times with different initial condition and the results were averaged.

The architecture of «bottle-neck» two-layer autoencoder and its learning algorithm are proposed. Such system is based on generalized neo-fuzzy neurons and is autoassociative “bottle-neck” modification of Kolmogorov’s neuro-fuzzy network. The proposed hybrid neo-fuzzy system of computational intelligence provides high quality of information compression, which are fed sequentially for processing. It is characterized by computational simplicity and high speed of the learning process.

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