Generalised Watson Distribution on the Hypersphere with Applications to Clustering

A family of probability density functions (pdfs) is defined on the unit hypersphere Sⁿ. The parameter space for the pdfs is G(d, n+ 1) × R_{≥ 0}, for 1 ≤ d≤ n, where G(d, n+ 1) is the Grassmannian of d-dimensional linear subspaces in Rⁿ⁺¹ and R_{≥ 0} is the range of values for a concentration parameter. This family of pdfs generalises the Watson distribution on the sphere S². It is shown that the pdfs are tractable, in that (i) a given pdf can be sampled efficiently, (ii) the parameters of a pdf can be estimated using maximum likelihood, and (iii) the Kullback–Leibler divergence and the Fisher–Rao metric on G(d, n+ 1) × R_{≥ 0} have simple forms. A wide range of shapes of the pdfs can be obtained by varying d and the concentration parameter. The pdfs are used to model clusters of feature vectors on the hypersphere. The clusters are compared using the Kullback–Leibler divergences of the associated pdfs. Experiments with the mnist, Human Activity Recognition and Gas Sensor Array Drift datasets show that good results can be obtained from clustering algorithms based on the Kullback–Leibler divergence, even if the dimension n of the hypersphere is high.