On metric spaces arising during formalization of problems of recognition and classification. Part 2: Density properties

Abstract

In order to obtain tractable formal descriptions of poorly formalized problems within the context of the algebraic approach to pattern recognition, we develop methods for analyzing metric configurations. In this paper, using the concepts of σ-isomorphism and σ-completion of metric configurations, a system of criteria for assessing the properties of “generalized density” is obtained. The analysis of the density properties along the axes of a metric configuration allowed us to formulate methods for calculating the topological neighborhoods of points and for finding the “grains” of metric condensations. The theoretical results point to a new plethora of algorithms for searching metric condensations − methods based on the “restoration” of the set (the condensation searched) using the data on the components of the projection of the set on the axes of the metric configuration. The only mandatory parameters of any algorithm of this family of algorithms are the metric itself and the distribution of σ, which characterizes the accuracy of the values of the metric.