Abstract
The problem of constructing metric ε-nets and corresponding coverings by balls for compact sets with a probability measure is considered. In the case of sets having metrically significant parts with a small measure (dark sets), methods for constructing ε-nets are combined with the deep holes method in a unified approach. According to this approach, a constructed metric net is supplemented with its deep hole (the most distant element of the set) until the required accuracy is achieved. An existing implementation of the method for a metric set with a given probability measure is based on a pure global search for deep holes. To construct dark coverings, the method is implemented on the basis of a random multistart. For the resulting nets, the logarithm of the number of their elements is shown to be close to ε-entropy, which means that they are optimal. Techniques for estimating the reliability and completeness of constructed (ε, δ)-coverings in the sense of C.E. Shannon are described. The methods under consideration can be used to construct coverings of implicitly given sets with a measure defined on the preimage and to recover compact supports of multidimensional random variables with an unknown distribution law.
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ACKNOWLEDGMENTS
This work was supported by the Russian Foundation for Basic Research, project no. 18-01-00465a.
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Translated by I. Ruzanova
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Kamenev, G.K. Method for Constructing Optimal Dark Coverings. Comput. Math. and Math. Phys. 58, 1040–1048 (2018). https://doi.org/10.1134/S0965542518070084
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DOI: https://doi.org/10.1134/S0965542518070084