Computation of Centroidal Voronoi Tessellations in High Dimensional Spaces

Owing to the natural interpretation and various desirable mathematical properties, centroidal Voronoi tessellations (CVTs) have found a wide range of applications and correspondingly a vast development in their literature. However, the computation of CVTs in higher dimensional spaces remains difficult. In this letter, we exploit the non-uniqueness of CVTs in higher dimensional spaces for their computation. We construct such high dimensional tessellations by decomposing into CVTs in one-dimensional spaces. We then prove that such a tessellation is centroidal under the condition of independence among densities over the 1-D spaces. Various numerical evaluations backup the theoretical result through the low energy of the grid-like tessellations, and are obtained with minimal computation time. We also compare the proposed decomposition method with the popular MacQueen's probabilistic method.