A Novel Algorithm for Clustering of Data on the Unit Sphere via Mixture Models

TL;DR

This paper introduces a new algorithm for clustering data on the unit sphere using mixture models, combining manifold optimization with a block successive lower-bound maximization framework, and demonstrates its effectiveness through simulations and applications.

Contribution

It proposes a novel maximum likelihood estimation algorithm for Kent mixture models that guarantees convergence and consistency, along with a BIC-like model selection criterion.

Findings

The algorithm converges and estimates are consistent.

The BIC-like criterion effectively determines the number of mixture components.

The method performs well in image segmentation and neural imaging applications.

Abstract

A new maximum approximate likelihood (ML) estimation algorithm for the mixture of Kent distribution is proposed. The new algorithm is constructed via the BSLM (block successive lower-bound maximization) framework and incorporates manifold optimization procedures within it. The BSLM algorithm is iterative and monotonically increases the approximate log-likelihood function in each step. Under mild regularity conditions, the BSLM algorithm is proved to be convergent and the approximate ML estimator is proved to be consistent. A Bayesian information criterion-like (BIC-like) model selection criterion is also derive, for the task of choosing the number of components in the mixture distribution. The approximate ML estimator and the BIC-like criterion are both demonstrated to be successful via simulation studies. A model-based clustering rule is proposed and also assessed favorably via simulations. Example applications of the developed methodology are provided via an image segmentation task and a neural imaging clustering problem.