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Efficient quantisation and weak covering of high dimensional cubes

Noonan, Jack and Zhigljavsky, Anatoly ORCID: 2022. Efficient quantisation and weak covering of high dimensional cubes. Discrete and Computational Geometry 68 , pp. 540-565. 10.1007/s00454-022-00396-7

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Let Zn={Z1,…,Zn} be a design; that is, a collection of n points Zj∈[−1,1]d. We study the quality of quantisation of [−1,1]d by the points of Zn and the problem of quality of coverage of [−1,1]d by Bd(Zn,r), the union of balls centred at Zj∈Zn. We concentrate on the cases where the dimension d is not small, d≥5, and n is not too large, n≤2d. We define the design Dn,δ as a 2d−1 design defined on vertices of the cube [−δ,δ]d, 0≤δ≤1. For this design, we derive a closed-form expression for the quantisation error and very accurate approximations for the coverage area vol([−1,1]d∩Bd(Zn,r)). We provide results of a large-scale numerical investigation confirming the accuracy of the developed approximations and the efficiency of the designs Dn,δ.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Publisher: Springer Verlag (Germany)
ISSN: 0179-5376
Funders: EPSRC
Date of First Compliant Deposit: 14 April 2022
Date of Acceptance: 2 February 2022
Last Modified: 22 May 2023 21:38

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