By Igor Aizenberg (auth.)

*Complex-Valued Neural Networks* have larger performance, examine speedier and generalize greater than their real-valued counterparts.

This e-book is dedicated to the *Multi-Valued Neuron* (MVN) and MVN-based neural networks. It features a accomplished statement of MVN concept, its studying, and functions. MVN is a complex-valued neuron whose inputs and output can be found at the unit circle. Its activation functionality is a functionality in basic terms of argument (phase) of the weighted sum. MVN derivative-free studying relies at the error-correction rule. A unmarried MVN can study these input/output mappings which are non-linearly separable within the genuine area. Such classical non-linearly separable difficulties as XOR and Parity *n* are the easiest that may be discovered through a unmarried MVN. one other very important benefit of MVN is a formal therapy of the section information.

These houses of MVN turn into much more impressive while this neuron is used as a uncomplicated one in neural networks. The *Multilayer Neural community in keeping with Multi-Valued Neurons* (MLMVN) is an MVN-based feedforward neural community. Its backpropagation studying set of rules is derivative-free and in response to the error-correction rule. It doesn't be afflicted by the neighborhood minima phenomenon. MLMVN outperforms many different laptop studying strategies when it comes to studying pace, community complexity and generalization power while fixing either benchmark and real-world category and prediction difficulties. one other attention-grabbing program of MVN is its use as a easy neuron in multi-state associative memories.

The ebook is addressed to these readers who increase theoretical basics of neural networks and use neural networks for fixing a number of real-world difficulties. it may even be very appropriate for Ph.D. and graduate scholars pursuing their levels in computational intelligence.

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**Sample text**

3. 2. Notice that function f1 ( x1 , x2 ) = x1 x2 may be obtained by changing the order of variables in function f 2 ( x1 , x2 ) = x1 x2 . It was shown in [6] that if some Boolean function is threshold, than any function obtained from the first one by the permutation of its variables is also threshold and its weighting vector can be obtained by the permutation of the weights in the weighting vector of the first function corresponding to the permutation of the variables. (a) a two layer neural network with two inputs, with one hidden layer containing two neurons, and the output layer containing a single neuron (b) a two layer neural network with two inputs, with one hidden layer containing two neurons, and the output layer containing a single neuron.

To implement the learning process, the backpropagation learning algorithm was suggested. A problem, which is necessary to solve, implementing the learning process for a feedforward neural network, is finding the hidden neurons errors. While the exact errors of output neurons can be easily calculated as the differences between the desired and actual outputs, for all the hidden neurons their desired outputs are unknown and therefore there is no straightforward way to calculate their errors. But without the errors it is not possible to adjust the weights.

26) s =1 E denotes MSE, N is the total number of samples (patterns) in the learning set and Es denotes the square error of the network for the sth pattern; where 30 1 Why We Need Complex-Valued Neural Networks? , N k =1 * ks 2 for N m output neu- rons. 26). 28) s 1 is used so as to simplify subsequent 2 derivations resulting from the minimization of E . 27) is a m is the output layer index, and the factor function of the weights. Indeed, it strictly depends on all the network weights. It is a principal assumption that the error depends not only on the weights of the neurons at the output layer, but on all neurons of the network.