Dimension-independent Markov chain Monte Carlo on the sphere

TL;DR

This paper develops dimension-independent Markov chain Monte Carlo methods for Bayesian analysis on high-dimensional spheres, leveraging Hilbert space techniques to efficiently sample from antipodally symmetric directional data posteriors.

Contribution

It introduces a novel approach to adapt linear space samplers to spherical domains using a push-forward construction, ensuring efficiency and spectral gap properties.

Findings

Algorithms demonstrate dimension-independent efficiency in experiments.

Methods successfully sample from posteriors with antipodally symmetric priors.

Approach extends linear space samplers to spherical settings with theoretical guarantees.

Abstract

We consider Bayesian analysis on high-dimensional spheres with angular central Gaussian priors. These priors model antipodally symmetric directional data, are easily defined in Hilbert spaces and occur, for instance, in Bayesian binary classification and level set inversion. In this paper we derive efficient Markov chain Monte Carlo methods for approximate sampling of posteriors with respect to these priors. Our approaches rely on lifting the sampling problem to the ambient Hilbert space and exploit existing dimension-independent samplers in linear spaces. By a push-forward Markov kernel construction we then obtain Markov chains on the sphere, which inherit reversibility and spectral gap properties from samplers in linear spaces. Moreover, our proposed algorithms show dimension-independent efficiency in numerical experiments.