A deep machine learning algorithm for construction of the Kolmogorov-Arnold representation

A. Polar[^1] and M. Poluektov[^2]

[1] Independent Software Consultant, Duluth, GA, USA

[2] International Institute for Nanocomposites Manufacturing, WMG, University of Warwick, Coventry CV4 7AL, UK 

Abstract

The Kolmogorov-Arnold representation is a proven adequate replacement of a continuous multivariate function by an hierarchical structure of multiple functions of one variable. The proven existence of such representation inspired many researchers to search for a practical way of its construction, since such model answers the needs of machine learning. This article shows that the Kolmogorov-Arnold representation is not only a composition of functions but also a particular case of a tree of the discrete Urysohn operators. The article introduces new, quick and computationally stable algorithm for constructing of such Urysohn trees. Besides continuous multivariate functions, the suggested algorithm covers the cases with quantised inputs and combination of quantised and continuous inputs. The article also contains multiple results of testing of the suggested algorithm on publicly available datasets, used also by other researchers for benchmarking. 

Keywords: deep machine learning, Kolmogorov-Arnold representation, discrete Urysohn operator, classification trees.