The Convergence Of The Multi-valued Neuronlearning Algorithm

The artificial neural network is a kind of mathematical model,which realizes information processing by applying structures similar to the neural synapse of brains and using methods of mathematics and physics etc.The artificial neural network connects a large number of simple neurons through different ways and determines the joint strength among elements by a certain learning process.Complex-valued neural network is one kind of the neural network,and the introduction of complex value makes the ability of a single neuron to learn the nonlinearly separable problem.The higher functionality of the complex-valued neural network makes the necessity for us to study on it.The multi-valued neuron is a neuron that has complex-valued weights and inputs/outputs,and both of its inputs and outputs are located on the unit circle.Its method is the theory of multi-valued threshold functions over the complex numbers field.The good effects of the multi-valued neural networks can be seen through current literatures,but we still worry about the learning algorithm without a derivation process and the convergence with no rigorous proofs.Then vulnerabilities of the convergence proof were found through derivations.Followed that,we focus on that how to improve the neuron,and whether the learing algorithm can be derived etc.Firstly,this paper gives the non-convergence theory of the traditional MVN learning algorithm in the 2-separable case after pointing out errors of the original convergence proof,and analyzes the existence and feasibility of our new theory.Secondly,the corresponding relation between inputs and outputs has been changed through twisting the unit circle of the multi-valued neuron so as to propose a new type of multi-valued neuron.This new MVN perfectly solved the non-convergence problem under the 2-separable case for the traditional one,and the convergence proof of the new MVN learning algorithm has been given in this paper.Thirdly,the multi-valued neuron learning algorithm has been deduced to realize the minimum error on the bases of gradient descent,and the convergence of that has been proved.