On the Theoretical Basis of Metric Analysis of Poorly Formalized Problems of Recognition and Classification

摘要

In many fields of modern science, there are problems adequate formalization of which is indis- pensable for obtaining practically and theoretically important results. In the terminology of the scientific school of academician Yu.I. Zhuravlev, a formalized problem is uniquely defined by the matrix of information and the information matrix. In the present paper, a whole class of issues related to the formalization of rec- ognition/classification problems is considered, and a universal formalism is proposed for carrying out a met- ric analysis of poorly formalized problems. Thus, the formalization of a problem can be represented as a suc- cessive transition from the set of original descriptions to a particular topology, then to a lattice, and then to a certain metric space. It is shown that the property of Zhuravlev's regularity is sufficient for the existence of bijective mappings between these mathematical constructs. The possibilities of application of the apparatus developed are illustrated by several issues important for the formalization of the problems: introduction of metrics on the sets of the features and metrics on the sets of objects and analysis of "interactions" between dissimilar feature descriptions.