Computing Voronoi adjacencies in high dimensional spaces by using linear programming

Abstract

Some algorithms in Pattern Recognition and Machine Learning as neighborhood-based classification and dataset condensation can be improved with the use of Voronoi tessellation. This paper shows the weakness of some existing algorithms of tessellation to deal with high-dimensional datasets. The use of linear programming can improve the tessellation procedures by focusing on Voronoi adjacency. It will be shown that the adjacency test based on linear programming is a version of the polytope search. However, the polytope search procedure provides more information than a simple Boolean test. This paper proposes a strategy to use the additional information contained in the basis of the linear programming algorithm to obtain other tests. The theoretical results are applied to tessellate several random datasets, and also for much-used datasets in Machine Learning repositories.