Minimum Density Hyperplanes

Associating distinct groups of objects (clusters) with contiguous regions of high probability density (high-density clusters), is a central assumption in statistical and machine learning approaches for the classification of unlabelled data. In unsupervised classification this cluster definition underlies a nonparametric approach known as density clustering. In semi-supervised classification, class boundaries are assumed to lie in regions of low density, which is equivalent to assuming that high-density clusters are associated with a single class. We propose a novel hyperplane classifier for unlabelled data that avoids splitting high-density clusters. The minimum density hyperplane minimises the integral of the empirical probability density function along a hyperplane. The link between this approach and density clustering is immediate. We are able to establish a link between the minimum density and the maximum margin hyperplanes, thus linking this approach to maximum margin clustering and semi-supervised support vector machine classifiers. We propose a globally convergent algorithm for the estimation of minimum density hyperplanes for unsupervised and semi-supervised classification. The performance of the proposed approach for unsupervised and semi-supervised classification is evaluated on a number of benchmark datasets and is shown to be very promising.