Geodesic
Convergence of an iterative algorithm for computing parameters of multi-valued threshold functions
Prikladnaâ diskretnaâ matematika, no. 1 (2018), pp. 107-115.

Voir la notice de l'article provenant de la source Math-Net.Ru

A $k$-valued threshold function is defined as $f(x_1,\dots,x_n)=i\in\{0,1,\dots,k-1\}\Leftrightarrow b_i\le L(x_1,\dots,x_n)$ 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 $\mathbb R$, $b_0$. 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.
Keywords: threshold functions, iterative algorithms
Mots-clés : convergence. 
@article{PDM_2018_1_a9,
     author = {A. V. Burdelev},
     title = {Convergence of an iterative algorithm for computing parameters of multi-valued threshold functions},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {107--115},
     publisher = {mathdoc},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2018_1_a9/}
}
TY  - JOUR
AU  - A. V. Burdelev
TI  - Convergence of an iterative algorithm for computing parameters of multi-valued threshold functions
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2018
SP  - 107
EP  - 115
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDM_2018_1_a9/
LA  - ru
ID  - PDM_2018_1_a9
ER  - 
%0 Journal Article
%A A. V. Burdelev
%T Convergence of an iterative algorithm for computing parameters of multi-valued threshold functions
%J Prikladnaâ diskretnaâ matematika
%D 2018
%P 107-115
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDM_2018_1_a9/
%G ru
%F PDM_2018_1_a9
A. V. Burdelev. Convergence of an iterative algorithm for computing parameters of multi-valued threshold functions. Prikladnaâ diskretnaâ matematika, no. 1 (2018), pp. 107-115. http://geodesic.mathdoc.fr/item/PDM_2018_1_a9/

[1] Burdelev A. V., Nikonov V. G., “A new algorithm for recognition of $k$-valued threshold functions”, Computational Nanotechnology, 2017, no. 1, 7–14 (in Russian)

[2] Obradovic Z., Parberry I., “Learning with discrete multi-valued neurons”, Proc. 7th Intern. Conf. Machine Learning (University of Texas, Austin, Texas, June 21–23, 1990), 392–399

[3] Minsky M., Papert S., Perceptrons, MIT Press, Cambridge, MA, 1969

[4] Nikonov V. G., Nikonov N. V., “Features of threshold representations of $k$-valued functions”, Tr. Diskr. Mat., 11, no. 1, 2008, 60–85 (in Russian)