Generalised Watson Distribution on the Hypersphere with Applications to Clustering

A family of probability density functions (pdfs) is defined on the unit hypersphere S-n. The parameter space for the pdfs is G(d, n + 1) x 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 S-2. 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) x 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.

Automated summary

This study defines a family of probability density functions on the unit hypersphere S-n and investigates their properties and parameter estimation methods. Various shapes of probability density functions can be obtained by adjusting the parameters. Experiments show that clustering algorithms based on Kullback-Leibler divergence can achieve good results even in high-dimensional scenarios.