Crystallization Learning with the Delaunay Triangulation

Abstract

Based on the Delaunay triangulation, we propose the crystallization learning to estimate the conditional expectation function in the framework of nonparametric regression. By conducting the crystallization search for the Delaunay simplices closest to the target point in a hierarchical way, the crystallization learning estimates the conditional expectation of the response by fitting a local linear model to the data points of the constructed Delaunay simplices. Instead of conducting the Delaunay triangulation for the entire feature space which would encounter enormous computational difficulty, our approach focuses only on the neighborhood of the target point and thus greatly expedites the estimation for high-dimensional cases. Because the volumes of Delaunay simplices are adaptive to the density of feature data points, our method selects neighbor data points uniformly in all directions and thus is more robust to the local geometric structure of the data than existing nonparametric regression methods. We develop the asymptotic properties of the crystallization learning and conduct numerical experiments on both synthetic and real data to demonstrate the advantages of our method in estimation of the conditional expectation function and prediction of the response.