Combinatorial analysis of the solvability properties of the problems of recognition and completeness of algorithmic models. Part 2: Metric approach within the framework of the theory of classification of feature values

Abstract

The properties of solvability/regularity of problems and correctness/completeness of algorithmic models are fundamental components of the algebraic approach to pattern recognition. In this paper, we formulate the principles of the metric approach to the data analysis of poorly formalized problems and hence with obtain metric forms of the criteria of solvability, regularity, correctness, and completeness. In particular, the analysis of the compactness properties of metric configurations allowed us to obtain a set of sufficient conditions for the existence of correct algorithms. These conditions can be used for assessment of the quality of the methods of formalization of the problems for arbitrary algorithms and algorithmic models. The general schema proposed for the data analysis of poorly formalized problems includes the criteria in the cross-validation form and can assess not only the quality of formalization, but also the extent of overtraining pertaining to the procedures of generation and selection of feature descriptions.