Non-lattice covering and quanitization of high dimensional sets

by   Jack Noonan, et al.

The main problem considered in this paper is construction and theoretical study of efficient n-point coverings of a d-dimensional cube [-1,1]^d. Targeted values of d are between 5 and 50; n can be in hundreds or thousands and the designs (collections of points) are nested. This paper is a continuation of our paper <cit.>, where we have theoretically investigated several simple schemes and numerically studied many more. In this paper, we extend the theoretical constructions of <cit.> for studying the designs which were found to be superior to the ones theoretically investigated in <cit.>. We also extend our constructions for new construction schemes which provide even better coverings (in the class of nested designs) than the ones numerically found in <cit.>. In view of a close connection of the problem of quantization to the problem of covering, we extend our theoretical approximations and practical recommendations to the problem of construction of efficient quantization designs in a cube [-1,1]^d. In the last section, we discuss the problems of covering and quantization in a d-dimensional simplex; practical significance of this problem has been communicated to the authors by Professor Michael Vrahatis, a co-editor of the present volume.


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