Han Cheng Lie, Daniel Rudolf, Björn Sprungk and I have just uploaded a preprint of our latest article, “Dimension-independent Markov chain Monte Carlo on the sphere”, to the arXiv. In this paper, motivated by problems such as Bayesian binary classification over continuous spaces, for which the parameter space is naturally an infinite-dimensional sphere of functions, we consider MCMC methods for inference on spheres of Hilbert spaces. In particular, we construct MCMC methods that have dimension-independent performance in terms of their acceptance probability, spectral gap, etc.; we also show how more naive approaches may lack basic properties such as Markovianity and reversibility; how how even sophisticated geometric MCMC approaches can still suffer from the curse of dimension.
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.