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A Recurrent Perceptron Learning Algorithm for Cellular Neural Networks

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A Recurrent Perceptron Learning Algorithm for Cellular Neural Networks ARI (1999) 51: 296}309 ( Springer-Verlag 1999 ORIGINAL ARTICLE C. GuK zelis, ' S. Karamamut ' I0 . Genc, A recurrent perceptron learning algorithm for cellular neural networks Received: 5 April 1999/Accepted: 11 May 1999 Abstract A supervised learning algorithm for obtaining nected only to the cells in their nearest neighborhood the template coe$cients in completely stable Cellular de"ned by the following metric: Neural Networks (CNNs) is analysed in the paper. The considered algorithm resembles the well-known percep- d(i, j; m(, j( )"max MDi!m( D, D j!j( DN tron learning algorithm and hence called as Recurrent Perceptron Learning Algorithm (RPLA) when applied to where (i, j) is the vector of integers indexing the cell C (i, j) a dynamical network. The RPLA learns pointwise de"ned in the ith row, jth column of the 2-dimensional array. The algebraic mappings from initial-state and input spaces system of equations describing a CNN with the neighbor- into steady-state output space; despite learning whole hood size of 1 is given in Eqs. 1}2. trajectories through desired equilibrium points. The R "! ) # + ) RPLA has been used for training CNNs to perform some xi,j A xi,j wk,l yi`k,j`l image processing tasks and found to be successful in k, l3M!1, 0, 1N binary image processing. The edge detection templates # + ) # found by RPLA have performances comparable to those zk,l ui`k,j`1 I (1) of Canny's edge detector for binary images. k, l3M!1, 0, 1N 1 ) ) " " ) MD # D!D ! DN Key words Cellular neural networks Learning yi,j f (xi,j): xi,j 1 xi,j 1 , (2) perceptron learning rule ) Image processing 2 3 where, A, I, wk,l and zk,l R are constant parameters. 3 3 ! 3 ! xi,j (t) R, yi,j (t) [ 1, 1], and ui,j [ 1, 1] respec- tively denotes the state, output, and (time-invariant) ex- 1 Introduction ternal input associated to a cell C (i, j). It is known in Chua and Yang (1988) that a CNN is A Cellular Neural Network (CNN) is a 2-dimensional completely stable if the feedback connection weights wk,l array of cells (Chua and Yang 1988). Each cell is made up are symmetric. Throughout the paper, the input connec- of a linear resistive summing input unit, and R-C linear tion weights zk,l are chosen to be symmetric for reducing dynamical unit, and a 3-region, symmetrical, piecewise- computational costs while the feedback connection linear resistive output unit. The cells in a CNN are con- weights wk,l are chosen symmetrically for ensuring com- " " " " plete stability, i.e., w~1,~1 w1,1 : a1, w~1,0 w1,0 : " " " " " a2, w~1,1 w1,~1 : a3, w0,~1 w0,1 : a4, w0,0 : a5; z "z :"b , z "z :"b , z "z :"b , ' ~1,~1 1,1 1 ~1,0 1,0 2 ~1,1 1,~1 3 C. GuK zelis, ( ) S. Karamahmut z "z :"b , z :"b . Hence, the number of con- Faculty of Electrical-Electronics Engineering, 0,~1 0,1 4 0,0 5 I0 stanbul Technical University, Istanbul, Maslak 80626, Turkey nection weights to be adapted is a small number, 11, for e-mail [email protected] the chosen neighborhood size of 1. So, the learning is Tel.: #90 212 285 3610, accomplished through modi"cation of the following Fax.: #90 212 285 3679 weight vector w3R11 whose entries are the feedback I0 . Genc, template coe$cients ai's the input template coe$cients Faculty of Engineering, Ondokuz May 18 University, bj's, and the threshold I. 55139, Samsun, Turkey " T T T " T e-mail: [email protected] w : [a b I] : [a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 I] . (3) 297 Several design methods and supervised learning algo- The lack of the derivative of error function prevents rithms for determining templates coe$cients of CNNs are using gradient-based methods for "nding template, min- proposed in the literature (Chua and Yang 1988; Vander- imizing the error. In order to overcome this problem, the berghe and Vandewalle 1989; Zou et al. 1990; Nossek et al. output function can be replaced (Karamahmut and 1992; Nossek 1996; Chua and Shi 1991; Chua and Thiran GuK zelis, 1994) with a continuously di!erentiable one 1991; Kozek et al. 1993; Schuler et al. 1992; Magnussen which is close to the original piecewise-linear function in and Nossek 1992; GuK zelis, 1992; Balsi 1992; Balsi 1993; Eq. 2. Whereas the gradient methods are now applicable, Schuler et al. 1993; Karamahmut and GuK zelis,, 1994; the error surfaces have almost #at regions resulting in GuK zelis, and Karamahumut 1994; Lu and Liu, 1998; Liu extremely slow convergence (Karamahmut and GuK zelis, 1997; Fajfar et al. 1998; ZaraH ndy 1999). As template design 1994). An alternative solution to this problem is to use methods, well-known relaxation methods for solving lin- methods not requiring the derivative of error. Such ear inequalities are used in Vanderberghe and Vandewalle a method is given in Kozek et al. (1993) by introducing (1989), Zou et al. (1990) for "nding one of the connection genetic optimization algorithms for the supervised learn- weights providing that the desired outputs are in ing of the optimal template coe$cients. The learning the equilibrium set of a considered CNN. However, for the algorithm analyzed in this paper, RPLA, constitutes an- methods in Vanderberghe and Vandewalle (1989), Zou other solution in this direction. The RPLA is, indeed, et al. (1990), there is not a general procedure on how to a reinforcement type learning algorithm: it terminates if specify an initial state vector yielding the desired output the output mismatching error is zero, otherwise it penaliz- for the given external inputs and the found weight vector. es connection weights in a manner similar to the percep- A trivial solution in the determination of such a proper tron learning rule. initial state vector is to take the desired output as the The RPLA is "rstly presented in (GuK zelis, and initial state; but this requires the knowledge of the desired Karamahmut 1994) for "nding template coe$cients of output which is not available for external inputs outside a completely-stable CNN to realise an input-(steady-state) the training set. On the other hand, a number of super- output map which is pointwise de"ned, i.e., described by vised learning algorithms to "nd connection weights of a set of training samples. Here, the input consists of two CNNs which yield the desired outputs for the given ex- parts: the "rst part is the external input and the second is ternal inputs and the predetermined initial states have the initial state. RPLA is a global learning type algorithm been developed in the past (Kozek et al. 1993; Schuler in the sense of Nossek (1996). This means that it aims to et al. 1992; Magnussen and Nossek 1992; GuK zelis,, 1992; learn not only equilibrium outputs but also their basins of Balsi 1992; Balsi 1993; Schuler et al. 1993; Karamahmut attraction. RPLA has been applied to nonlinear B-tem- and GuK selis, 1994; GuK zelis, and Karamahmut 1994). (see plate CNNs (Yalc,mn and GuK zelis, 1996) as well as linear Nossek (1996) for a review.) The backpropagation B-template CNNs; moreover a modi"ed version of it has through time algorithm is applied in Schuler et al. (1992) been used for learning regularization parameters in CNN- for learning the desired trajectories in continuous-time based early vision models (GuK zelis, and GuK nsel 1995; GuK n- CNNs. A modi"ed alternating variable method is used in sel and GuK zelis, 1995). Magnussen and Nossek (1992) for learning steady-state This paper is concerned with the convergence proper- outputs in discrete-time CNNs. Both of these algorithms ties of RPLA as well as its performance in learning image are proposed for any kind of CNNs since they do not processing tasks. It is shown in the paper that RPLA with impose any constraint needed to be imposed on connec- asu$ciently small constant learning rate converges, in tion weights for ensuring complete stability and the bi- "nite number steps, to a solution weight vector if such polarity of steady-state outputs. It is described in GuK zelis, a solution exits and if the conditions of Theorem 3 are (1992) that the supervised learning of steady-state outputs satis"ed. The RPLA is indeed reduced to the perceptron in completely stable generalized CNNs (GuK zelis, and Chua learning rule (Rosenblatt 1962) if feedback template coe$- 1993) is a constrained optimization problem, where the cients (except for the self-feedback one) are set to zero, i.e., objective function is the output error function and con- the corresponding CNN is in the linear threshold class straints are due to some qualitative and quantitative (Chua and Shi 1991). This means that a CNN trained with design requirements such as the bipolarity of this steady- an RPLA for a su$ciently small constant learning rate is ) state outputs and complete stability. The recurrent capable of learning any locally de"ned function Flocal ( ) backpropagation algorithm (Pineda 1988) is applied in :[!1, 1]9 P M!1, 1N of the external input whenever its Balsi (1992) and Balsi (1993) to a modi"ed version of CNN domain space speci"ed by a 3]3 nearest neighborhood, is di!ering from the original CNN model in the following linearly separable. respects: 1) cells are fully-connected, 2) the output function The structure of the paper is as follows.
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