# Complex-Valued Neural Networks with Multi-Valued Neurons by Igor Aizenberg (auth.)

By Igor Aizenberg (auth.)

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

This publication is dedicated to the *Multi-Valued Neuron* (MVN) and MVN-based neural networks. It includes a entire remark of MVN concept, its studying, and purposes. MVN is a complex-valued neuron whose inputs and output can be found at the unit circle. Its activation functionality is a functionality merely of argument (phase) of the weighted sum. MVN derivative-free studying relies at the error-correction rule. A unmarried MVN can examine 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 best that may be discovered through a unmarried MVN. one other very important benefit of MVN is a formal therapy of the part information.

These homes of MVN develop into much more striking whilst this neuron is used as a uncomplicated one in neural networks. The *Multilayer Neural community in line with Multi-Valued Neurons* (MLMVN) is an MVN-based feedforward neural community. Its backpropagation studying set of rules is derivative-free and in line with the error-correction rule. It doesn't be afflicted by the neighborhood minima phenomenon. MLMVN outperforms many different computer studying innovations by way of studying pace, community complexity and generalization strength whilst fixing either benchmark and real-world class and prediction difficulties. one other attention-grabbing program of MVN is its use as a simple neuron in multi-state associative memories.

The booklet is addressed to these readers who enhance theoretical basics of neural networks and use neural networks for fixing quite a few real-world difficulties. it's going to even be very appropriate for Ph.D. and graduate scholars pursuing their levels in computational intelligence.

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**Additional info for Complex-Valued Neural Networks with Multi-Valued Neurons **

**Sample text**

If we treate the phase as just real numbers belonging to the interval [0, 2π [ or [ −π , π [ , then we make a great mistake, be- cause in this way the physical nature of the phase is completely eliminated. 3) Since the functionality of a single complex-valued neuron is higher than the one of a single real-valued neuron, the functionality of complex-valued neural networks is also higher than the functionality of their real-valued counterparts. A smaller complex-valued neural network can learn faster and generalize better than a real-valued neural network.

10), in order to calculate the errors of all other neurons. The heuristic idea is to share the errors of output neurons, which can be calculated because their desired outputs are known (unlike the ones of the hidden neurons), with all the hidden neurons. Basically, the entire learning process consists of two passes through all the different layers of the network: a forward pass and a backward pass. In the forward pass, the inputs are propagated from the input layer of the network to the first hidden layer and then, layer by layer, output signals from the hidden neurons are propagated to the corresponding inputs of the following layer neurons.

There- fore, the OR function can be implemented with the threshold neuron using the weighting vector ~ W = (−1,3,3) obtained as the result of the learning process. 20 1 Why We Need Complex-Valued Neural Networks? 3. Let us learn the function f ( x 1 , x2 ) = x1 & x2 ( x means the ne# 1) 2) 3) 4) x1 x2 1 1 -1 -1 1 -1 1 -1 W = (1,1,1) gation of the Boolean variable x, in the alphabet {1, -1} x = − x ) using the threshold neuron. 4 shows the function values and the entire -1 learning set containing four input vectors and four 1 values of the function, respectively.