Non-lattice covering and quanitization of high dimensional sets

TL;DR

This paper advances the theory and construction of efficient, nested point coverings and quantization schemes for high-dimensional cubes, improving upon previous designs and extending methods to simplices.

Contribution

It develops new theoretical constructions for nested coverings and quantization schemes in high dimensions, surpassing previous designs and extending to simplices.

Findings

New constructions provide better coverings than previous methods.

Extended theoretical analysis for quantization in high-dimensional cubes.

Practical recommendations for designing efficient high-dimensional coverings.

Abstract

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 \cite{us}, where we have theoretically investigated several simple schemes and numerically studied many more. In this paper, we extend the theoretical constructions of \cite{us} for studying the designs which were found to be superior to the ones theoretically investigated in \cite{us}. 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 \cite{us}. 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.