Geodesic slice sampling on the sphere

2023-01-19

AI Summary

This work addresses efficient sampling from probability distributions on the spherical manifold. We propose the first geodesic slice sampling Markov chain method specifically designed for the sphere, comprising both a practical, parameter-free, dimension-agnostic contraction variant and a theoretically rigorous ideal version. Our approach constitutes the first extension of slice sampling to non-Euclidean manifolds, leveraging spherical geometry and geodesic dynamics to naturally enforce support constraints. Under mild regularity conditions, we establish uniform ergodicity for the contraction variant. Experiments on Bingham distributions and von Mises–Fisher mixture models demonstrate that our method significantly outperforms random-walk Metropolis–Hastings and Hamiltonian Monte Carlo in mixing efficiency. It thus provides a robust, adaptive, and geometrically aware tool for directional data modeling and shape analysis.

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 in any dimension 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 the Bingham distribution 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.

Problem

• Efficient sampling of distributions on the sphere
• Designing geodesic slice sampling Markov chains
• Outperforming standard samplers like Metropolis-Hastings

Innovation

• Geodesic slice sampling on sphere
• Shrinkage-based algorithm without tuning
• Efficient high-dimensional spherical distribution sampling