By Igor Aizenberg (auth.)
Complex-Valued Neural Networks have greater performance, study 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 encompasses a entire statement of MVN conception, 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 in basic terms of argument (phase) of the weighted sum. MVN derivative-free studying is predicated 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 best that may be discovered via 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 outstanding whilst this neuron is used as a uncomplicated one in neural networks. The Multilayer Neural community in response to 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 velocity, community complexity and generalization potential while fixing either benchmark and real-world category and prediction difficulties. one other fascinating program of MVN is its use as a simple neuron in multi-state associative memories.
The publication is addressed to these readers who boost 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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Extra info for Complex-Valued Neural Networks with Multi-Valued Neurons
7a). This network contains the input layer, which distributes the input signals x1 and x 2 , one hidden layer containing Neurons 1 and 2 and one output layer containing a single Neuron 3. This is the simplest possible non-trivial multilayer feedforward neural network (MLF). It is the simplest possible network because it contains a minimum amount of layers and neurons to be non-trivial (two layers including one hidden layer and one output layer, two neurons in the hidden layer, and one neuron in the output layer).
It is well known that a multilayer feedforward neural network is a universal approximator (for the first time this was clearly proven in  and ). However, the Kolmogorov’s Theorem, being very important, is a typical “existence theorem”. It justifies only the existence of the solution. , 2n + 1 . , xn ) , which we want to approximate. 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.
17): w 0 = 1 − 2 = −1; w1 = 1 + (−2) ⋅1 = −1; w 2 = 1 + (−2) ⋅ (−1) = 3 . ~ Thus, W = (−1,−1,3) . The weighted sum after the correction is equal to z = −1 + (−1) ⋅ 1 + 3 ⋅ (−1) = −5 ; ϕ ( z ) = sgn( z ) = sgn(−5) = −1 . Since f (1, −1) = −1 , no further correction of the weights is needed. 3) Inputs (-1, 1). The weighted sum is equal to ϕ ( z ) = sgn( z ) = sgn(3) = 1 . Since z = −1 + (−1) ⋅ (−1) + 3 ⋅ 1 = 3 ; f (−1,1) = −1 , we have to correct the weights. According to (17) δ = −1 − 1 = −2 .