Abstract
A compact metric space with a bounded Borel measure is considered. Any measurable set of diameter not exceeding r is called r-cluster. The existence of a collection consisting of a fixed number of 2r-clusters possessing the following properties is investigated: the clusters are located at the distance r from each other and the collection measure (the total measure of the clusters in the collection) is close to the measure of the entire space. It is proved that there exists a collection with a maximum measure among such collections. The concept of r-parametric discretization of the distribution of distances into short, medium, and long distances is defined. In terms of this discretization, a lower bound on the measure of the maximum-measure collection is obtained.
Similar content being viewed by others
References
Yu. I. Zhuravlev and V. V. Nikiforov, “Recognition algorithms based on estimate evaluation,” Kibernetika, No. 3, 1–11 (1971).
M. A. Aizerman, E. M. Bravermann, and L. I. Rozonoer, The Metod of Potential Functions in Machine Learning (Nauka, Moscow, 1970) [in Russian].
M. E. Celebi, H. A. Kingravi, and P. A. Vela, “A comparative study of efficient initialization methods for the k-means clustering algorithm,” Expert Syst. Appl. 40 (1), 200–210 (2013).
R. C. De Amorim and B. Mirkin, “Minkowski metric, feature weighting and anomalous cluster initializing in k-means clustering,” Pattern Recogn. 45, 1061–1075 (2012).
C. C. Aggarwal and C. K. Reddy, Data Clustering: Algorithms and Applications (CRC, 2013).
N. G. Zagoruiko, “The hypothesis of compactness and -compactness in data analysis methods,” Sib. Zh. Ind. Mat. 1 (1), 114–126 (1998).
E. M. Bravermann, “Experiments on learning a machine to image recognition,” Avtom. Telemekh. 23, 349–365 (1962).
M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces (Springer, 2007).
A. M. Vershik, “Universal Uryson space, Gromov’s metric triples, and random metrics on the natural series,” Usp. Mat. Nauk 53 (5 (323)), 57–64 (1998).
E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis (Fizmatlit, Moscow, 2004) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.S. Pushnyakov, 2018, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2018, Vol. 58, No. 4, pp. 626–635.
Rights and permissions
About this article
Cite this article
Pushnyakov, A.S. On The Relationships of Cluster Measures and Distributions of Distances in Compact Metric Spaces. Comput. Math. and Math. Phys. 58, 612–620 (2018). https://doi.org/10.1134/S0965542518040140
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542518040140