Informatics and Applications
2026, Volume 20, Issue 1, pp 39-44
METHODS FOR GENERATING METRICS ON OBJECT SETS IN THE CONTEXT OF THEORY OF TOPOLOGICAL DATA ANALYSIS. PART 1. METRICS BASED ON DISTANCES BETWEEN FEATURE VALUES
Abstract
Distance metrics on object sets are widely used in various machine learning algorithms. However, generating such metrics for specific application tasks is a nontrivial challenge. Typically, researchers are limited to selecting from established empirical metrics and, in some cases, fine-tuning their parameters. The paper proposes several theoretical approaches developed within the context of topological data analysis. These approaches include metrics derived from distances between feature values, analysis of multidimensional spaces, the implications of Urysohn's embedding theorem, and the introduction of a lattice of feature combinations.
The proposed framework enables the systematic generation of problem-oriented metrics on object sets  .
The paper presents a rigorous analysis of the theoretical transition from metrics on feature value sets  to  . Experimental validation of the various branches of the proposed formalism is deferred to future publications.
[+] References (14)
- Zhuravlev, Yu. I. 1977. Correct algebras over sets of incorrect (heuristic) algorithms. I. Cybernetics 13(4):483–494.
- Zhuravlev, Yu. I. 1978. Ob algebraicheskom podkhode k resheniyu zadach raspoznavaniya i klassifikatsii [On the algebraic approach to solving recognition and classification
tasks].Problemy kibernetiki [Problems of Cybernetics] 33:5–68.
- Torshin, I. Yu., and K. V. Rudakov. 2015. On the theoretical
basis of metric analysis of poorly formalized
problems of recognition and classification. Pattern Recognition Image Analysis 25(4):577–587. doi:
10.1134/S1054661815040252. EDN: UWEXTB.
- Torshin, I. Yu. 2024. O porozhdenii sinteticheskikh priznakov
na osnove opornykh tsepey i proizvol’nykh metrik
v ramkakh topologicheskogo podkhoda k analizu dannykh.
Chast’ 1. Vklyuchenie v formalizm empiricheskikh
funktsiy rasstoyaniya [On the generation of synthetic features
based on support chains and arbitrarymetrics within
a topological approach to data analysis. Part 1. Inclusion
of empirical distance functions into the formalism]. Informatika
i ee Primeneniya—Inform. Appl. 18(1):71–77. doi:
10.14357/19922264240110. EDN: RIVOXR.
- Torshin, I. Yu. 2025. Metrizatsiya diskretnykh topologicheskikh
prostranstv v kontekste teorii reshetok. Chast’ 2.
Prakticheskiy analiz sledstviy teoremy o regulyarnosti
i normal’nosti [Metrization of discrete topological spaces
in the context of lattice theory. Part 2. Practical analysis
of the consequences of the theorem on regularity
and normality]. Informatika i ee Primeneniya — Inform.
Appl. 19(2):55–62. doi: 10.14357/19922264250207.EDN:
JTNKFQ.
- Torshin, I. Yu. 2025. Metrizatsiya diskretnykh topologicheskikh
prostranstv v kontekste teorii reshetok. Chast’ 1.
O normal’nosti prostranstv [Metrization of discrete topological
spaces in the context of lattice theory.Part 1.On the
normality of spaces]. Informatika i ee primeneniya — Inform.
Appl. 19(1):82–88. doi: 10.14357/19922264250111.
EDN: CAWKMO.
- Rudakov, K. V. 1988. Symmetric and functional constraints
for classification algorithms. Soviet Mathematics
Doklady 36(3):428–431.
- Hastie, T., R. Tibshirani, and J.H. Friedman. 2009.
The elements of statistical learning: Data mining, inference,
and prediction. New York, NY: Springer. 745 p. doi:
10.1007/978-0-387-84858-7.
- Aleksandrov, P. S., and A. N. Kolmogorov. 1948. Vvedenie
v obshchuyu teoriyu mnozhestv i funktsiy [Introduction
to the general theory of sets and functions]. Moscow:
GITTL. 308 p.
- Munkres, J.R. 2000. Topology. 2nd ed. Upper Saddle
River, NJ: Prentice Hall, Inc. 537 p.
- Birkhoff, G. 1967. Lattice theory. 3rd ed. AMS colloquium
publications ser. Providence, RI: AmericanMathematical
Society. Vol. 25. 418 p.
- Deza, E. I., and M. M. Deza. 2006. Dictionary of distances.
Amsterdam, Netherlands: North-Holland, Elsevier. 412 p.
doi: 10.1016/B978-0-44452087-6.X5000-8.
- Kolmogorov, A. N., and S. V. Fomin. 1989. Elementy teorii
funktsiy i funktsional’nogo analiza [Elements of the theory
of functions and functional analysis]. 6th ed. Moscow:
Nauka. 624 p.
- Vorontsov, K. V. 2010. Kombinatornaya teoriya nadezhnosti
obucheniya po pretsedentam [Combinatorial theory
of reliability of precedents-based learning]. D.Sc. Diss.
Moscow: VC RAN. 271 p.
[+] About this article
Title
METHODS FOR GENERATING METRICS ON OBJECT SETS IN THE CONTEXT OF THEORY OF TOPOLOGICAL DATA ANALYSIS. PART 1. METRICS BASED ON DISTANCES BETWEEN FEATURE VALUES
Journal
Informatics and Applications
2026, Volume 20, Issue 1, pp 39-44
Cover Date
2026-01-04
DOI
10.14357/19922264260105
Print ISSN
1992-2264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
topological data analysis; distance functions on objects; algebraic approach to algorithm design; theory of feature value analysis
Authors
I. Yu. Torshin 
Author Affiliations
Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
|