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(21.15)
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The vector s has the properties s2 = 2 and e s = 1. Note that if = 0, Equation (21.15) becomes the equation of a point (Equation 21.10). The same result can be obtained by first computing the volume of s = e x1 x2 x3 xn+1
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and then taking the dual of the latter with the pseudoscalar I = s = s I
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Let us give as illustration the computation of a plane and a sphere of positions: = e x1 x2 x3 s = s I = e x1 x2 x3 x4 I = p 1 2
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(21.18)
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3. Real-valued Neural Networks
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The approximation of nonlinear mappings using neural networks is useful in various aspects of signal processing, such as in pattern classification, prediction, system modeling and identification. This section reviews the fundamentals of standard real-valued feed-forward architectures. Cybenko [4] used, for the approximation of a continuous function, g x , the superposition of weighted functions:
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where is a continuous discriminatory function like a sigmoid, wj and x j wj n . Finite sums of the form of Equation (21.19) are dense in C 0 In , if gk x yk x < for a given > 0 and all x 0 1 n . This is called the density theorem and is a fundamental concept in approximation theory and nonlinear system modeling [4,9]. A structure with k outputs yk , having several layers using logistic functions, is known as a Multilayer Perceptron (MLP) [10]. The output of any neuron of a hidden layer or of the output layer can be represented in a similar way, oj = f j
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(21.20)
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where fj is logistic and fk is logistic or linear. Linear functions at the outputs are often used for pattern classification. In some tasks of pattern classification, a hidden layer is necessary, whereas in some tasks of automatic control, two hidden layers may be required. Hornik [9] showed that standard multilayer feed-forward networks are able to accurately approximate any measurable function to a desired degree. Thus, they can be seen as universal approximators. In the case of a training failure, we should attribute any error to inadequate learning, an incorrect number of hidden neurons, or a poorly defined deterministic relationship between the input and output patterns. Poggio and Girosi [11] developed the Radial Basis Function (RBF) network, which consists of a superposition of weighted Gaussian functions, yj x = wji Gi Di x ti (21.21)
where yj is the j-output, wij Gi is a Gaussian function, Di an N N dilatation diagonal matrix, and x ti n . The vector ti is a translation vector. This architecture is supported by the regularization theory.
4. Complex MLP and Quaternionic MLP
An MLP is defined to be in the complex domain when its weights, activation function and outputs are complex-valued. The selection of the activation function is not a trivial matter. For example, the extension of the sigmoid function from to C, f z = 1 1 + e z (21.22)
where z C, is not allowed, because this function is analytic and unbounded [12]; this is also true for 2 the functions tanh(z) and e z . We believe these kinds of activation function exhibit problems with convergence in training due to their singularities. The necessary conditions that a complex activation f z = a x y + ib x y has to fulfill are: f z must be nonlinear in x and y, the partial derivatives ax ay bx and by must exist, ax by = bx ay , and f z must not be entire. Accordingly, Georgiou and Koutsougeras [12] proposed the formulation: f z = z 1 c+ z r (21.23)
where c r + . These authors thus extended the traditional real-valued back-propagation learning rule to the complex-valued rule of the Complex Multilayer Perceptron (CMLP). Arena et al. [13] introduced the Quaternionic Multilayer Perceptron (QMLP), which is an extension of the CMLP. The weights, activation functions and outputs of this net are represented in terms of quaternions [14]. Arena et al. chose the following nonanalytic bounded function: f q = f q0 + q1i + q2j + q3k = 1 1 + e q0 + 1 1 + e q1 i+ 1 1 + e q2 j+ 1 1 + e q3 k (21.24)
where f is now the function for quaternions. These authors proved that the superposition of such functions accurately approximates any continuous quaternionic function defined in the unit polydisc of C n . The extension of the training rule to the CMLP was demonstrated in [13].