Geodesic slice sampling on the sphere

TL;DR

This paper introduces efficient geodesic slice sampling algorithms for distributions on the sphere, demonstrating superior mixing and performance over traditional methods in complex directional data applications.

Contribution

It proposes a shrinkage-based and an idealized geodesic slice sampling method that are efficient, parameter-free, and proven to be uniformly ergodic under weak conditions.

Findings

Achieves excellent mixing on challenging distributions

Outperforms standard samplers like Metropolis-Hastings and HMC

Runs efficiently without tuning parameters

Abstract

Probability measures on the sphere form an important class of statistical models and are used, for example, in modeling directional data or shapes. Due to their widespread use, but also as an algorithmic building block, efficient sampling of distributions on the sphere is highly desirable. We propose a shrinkage based and an idealized geodesic slice sampling Markov chain, designed to generate approximate samples from distributions on the sphere. In particular, the shrinkage-based version of the algorithm can be implemented such that it runs efficiently and has no tuning parameters. We verify reversibility and prove that under weak regularity conditions geodesic slice sampling is uniformly ergodic. Numerical experiments show that the proposed slice samplers achieve excellent mixing on challenging targets including distributions arising in rigid-registration problems and mixtures of von Mises-Fisher distributions. In these settings our approach outperforms standard samplers such as random-walk Metropolis-Hastings and Hamiltonian Monte Carlo.