Probabilistic quantum clustering

Abstract

Quantum Clustering is a powerful method to detect clusters with complex shapes. However, it is very sensitive to a length parameter that controls the shape of the Gaussian kernel associated with a wave function, which is employed in the Schrödinger equation with the role of a density estimator. In addition, linking data points into clusters requires local estimates of covariance which requires further parameters. This paper proposes a Bayesian framework that provides an objective measure of goodness-of-fit to the data, to optimise the adjustable parameters. This also quantifies the probabilities of cluster membership, thus partitioning the data into a specific number of clusters, where each cluster probability is estimated through an aggregated density function composed of the data samples that generate the cluster, having each cluster an associated probability density function P(K|X); this probability can be used as a measure of how well the clusters fit the data. Another main contribution of the work is the adaptation of the Schrödinger equation to deal with local length parameters for cluster discrimination by density. The proposed framework is tested on real and synthetic data sets, assessing its validity by measuring concordance with the Jaccard score.