Finding weight vectors of neurons over the field of complex numbers (Q2730544)

This article deals with a neuron model. Let \(X=(-1,x_1,\ldots, x_{n})\) be an input signals vector of the neuron and let \(W=(w_0,w_1,\ldots,w_{n})\) be a weight vector of the neuron, where \(x_{i}\in \{1,-1\}\), \(i=1,\ldots,n\), and \(w_{i},\;i=0,\ldots,n\), are complex numbers. Denote \(S=\sum\limits_{i=0}^{n}x_{i}w_{i}\) and let the complex plane be divided onto \(k\) equal sectors, where \(k\) is an even number. The author defines the predicate NEWLINE\[NEWLINEP(S)=\begin{cases}\;1,& \text{for \(S\) in a sector with even index;}\\ -1,& \text{for \(S\) in a sector with odd index}.\end{cases}NEWLINE\]NEWLINE The value of \(P(S)\) is the output of the neuron. The author proposes an algorithm of finding a weight vector \(W\) such that for given inputs \(X_{i}\) we obtain outputs \(f_{i}=P(X_{i}\cdot W)\).