Computation of Centroidal Voronoi Tessellations in High Dimensional spaces

TL;DR

This paper introduces a novel method for computing centroidal Voronoi tessellations in high-dimensional spaces by constructing them from one-dimensional CVTs, leveraging their non-uniqueness and independence conditions.

Contribution

It presents a new approach to efficiently compute high-dimensional CVTs using one-dimensional tessellations, supported by theoretical proofs and numerical evaluations.

Findings

High-dimensional CVTs can be constructed from 1D CVTs under independence conditions.

The proposed method produces low-energy, grid-like tessellations efficiently.

Numerical results confirm the theoretical advantages of the approach.

Abstract

Owing to the natural interpretation and various desirable mathematical properties, centroidal Voronoi tessellations (CVT) have found a wide range of applications and correspondingly a vast development in their literature. However the computation of CVT in higher dimensional spaces still remains difficult. In this paper, we exploit the non-uniqueness of CVTs in higher dimensional spaces for their computation. We construct such high dimensional tessellations from CVTs in one-dimensional spaces. We then prove that such a tessellation is centroidal under the condition of independence among densities over the one-dimensional spaces considered. Various numerical evaluations backup the theoretical result through the low energy of the tessellations. The resulting grid-like tessellations are obtained efficiently with minimal computation time.