A simple geometric method for navigating the energy landscape of centroidal Voronoi tessellations

Abstract

Finding optimal (or low energy) centroidal Voronoi tessellations (CVTs) on a 2D domain is a challenging problem. One must navigate an energy landscape whose desirable critical points have sufficiently small basins of attractions that they are inaccessible with Monte-Carlo initialized gradient descent methods. We present a simple deterministic method for efficiently navigating the energy landscape in order to it access these low energy CVTs. The method has two parameters and is based upon each generator moving away from the closest neighbor by a certain distance. We give a statistical analysis of the performance of this hybrid method comparing with the results of a large number of runs for both Lloyd's method and state of the art quasi-Newton methods. Stochastic alternatives are also considered.