Convergence of an iterative algorithm for computing parameters of multi-valued threshold functions

A $k$-valued threshold function is defined as $f(x_1,\dots,x_n)=i\in\{0,1,\dots,k-1\}\Leftrightarrow b_i\le L(x_1,\dots,x_n)<b_{i+1}$ where $L(x_1,\dots,x_n)=a_1x_1+a_2x_2+\dots+a_nx_n$ is a linear form in variables $x_1,\dots,x_n$ with the values in $\{0,1,\dots,k-1\}$ and coefficients $a_1,\dots,a_n$ in $\mathbb R$ and $b_0,\dots,b_k$ are some thresholds for $L$ in $\mathbb R$, $b_0<b_1<\dots<b_k$. A. V. Burdelev and V. G. Nikonov have created and published in J. Computational Nanotechnology (2017, no. 1, pp. 7–14) an iterative algorithm for computing coefficients $a_1,\dots,a_n$ and thresholds $b_0,\dots,b_k$ for any $k$-valued threshold function $f(x_1,\dots,x_n)$ given by its values $f(c_1,\dots,c_n)$ for all $(c_1\dots c_n)$ in $\{0,\dots,k-1\}^n$. In computer experiment they showed the convergence of this algorithm on many different examples. Here, we present a theoretical proof of this algorithm convergence on each $k$-valued threshold function for a finite number of steps (iterations). The proof is very much similar to the geometrical proof of perceptron convergence theorem by M. Minsky and S. Papert.