10 research outputs found
Efficient Random Walks on Riemannian Manifolds
According to a version of Donsker's theorem, geodesic random walks on
Riemannian manifolds converge to the respective Brownian motion. From a
computational perspective, however, evaluating geodesics can be quite costly.
We therefore introduce approximate geodesic random walks based on the concept
of retractions. We show that these approximate walks converge in distribution
to the correct Brownian motion as long as the geodesic equation is approximated
up to second order. As a result we obtain an efficient algorithm for sampling
Brownian motion on compact Riemannian manifolds.Comment: 14 pages; v3: published versio
Dimension‐independent Markov chain Monte Carlo on the sphere
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 density estimation and binary 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