yx =

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wj f wj x + wj x +

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(21.32)

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The extension of the MLP is straightforward. The equations using the geometric product for the outputs of hidden and output layers are given by: oj = f yk = f k

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j=1 Ni i=1 Nj

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wji xji + wji xji +

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wkj okj + wkj okj +

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(21.33)

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In radial basis function networks, the dilatation operation, given by the diagonal matrix Di , can be i i implemented by means of the geometric product with a dilation Di = e 2 [7], i.e., Di x ti Di x ti Di

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(21.34) (21.35)

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yk x =

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wkj Gj Dj xji tj Dj

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Note that in the case of the geometric RBF we are also using an activation function according to Equation (21.25).

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Learning Rule

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6. Learning Rule

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This section demonstrates the multidimensional generalization of the gradient descent learning rule in geometric algebra. This rule can be used for training the Geometric MLP (GMLP) and for tuning the weights of the Geometric RBF (GRBF). Previous learning rules for the real-valued MLP, complex MLP [12], and quaternionic MLP [13] are special cases of this extended rule.

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6.1 Multidimensional Back-propagation Training Rule

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The norm of a multivector x for the learning rule is given by: x = xx

1 2 1 2

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(21.36)

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The geometric neural network with n inputs and m outputs approximates the target mapping function: yt

Gp q r

Gp q r

(21.37)

where Gp q r is the n-dimensional module over the geometric algebra Gp q r [15]. The error at the output of the net is measured according to the metric: E= 1 2 yw yt

(21.38)

where X is some compact subset of the Clifford module Gp q r involving the product topology derived from Equation (21.36) for the norm, and where yw and yt are the learned and target mapping functions, respectively. The back-propagation algorithm [10] is a procedure for updating the weights and biases. This algorithm is a function of the negative derivative of the error function (Equation (21.38)) with respect to the weights and biases themselves. The computing of this procedure is straightforward, and here we will only give the main results. The updating equation for the multivector weights of any hidden j-layer is wij t + 1 =

Nk kj k

wkj

F net ij

Oi + wij t

(21.39)

for any k-output with a nonlinear activation function wjk t + 1 = ykt yka F net jk oj + wjk t (21.40)

and for any k-output with a linear activation function wjk t + 1 = ykt yka oj + wjk t (21.41)

In the above equations, F is the activation function defined in Equation (21.25), t is the update step, and are the learning rate and the momentum, respectively, is the Clifford or geometric product, is the scalar product, and is the multivector anti-involution (reversion or conjugation). In the case of the non-Euclidean x G0 3 0 , corresponds to the simple conjugation. Each neuron now consists of p + q + r units, each for a multivector component. The biases are also multivectors and are absorbed as usual in the sum of the activation signal, here defined as net ij . In the learning rules, Equations (21.39) (21.41), the computation of the geometric product and the anti-involution varies depending on the geometric algebra being used [18]. To illustrate this, the conjugation required in the learning rule for quaternion algebra is x = x0 x1 1 x2 2 x3 1 2 , where x G0 2 0 .

Clifford-valued NNs in Pattern Classification

W31 X1 1 W41

4 X2 2 W51 5

Figure 21.5 Geometric MLP for training using genetic algorithms. The weights are Clifford numbers.

6.2 Geometric Learning Using Genetic Algorithms

This learning method utilizes a standard genetic algorithm (see [19]). All Clifford weights and biases of the geometric MLP depicted in Figure 21.5 are encoded as a genotype. Note that the weights are Clifford numbers which have a length of 2n , according to the total dimension n of the Clifford algebra Gn . The used objective function is E w31 w32 w13 = 1 2

oi oi

(21.42)

where wij stands for the Clifford weights and oi and oi for the actual and target outputs respectively. As usual, in each iteration the standard genetic operations (mutation, crossover) are carried out and the fitness of the population is evaluated. The procedure carries on and on until little progress is noticed. The qualified genotype guarantees a good performance of the neural net. Interestingly enough, this procedure takes into account the involved Clifford product. We trained a geometric MLP using the genetic algorithm for the problem of pattern classification. For this case, we utilized as preprocessing the hypercomplex moments. As expected, this kind of training proved not to be trapped in a local minimum. This experiment is discussed in detail in Section 9.