This 2023 work on Slice sampler on manifolds, as presented in the algorithms seminar in Warwick during a recent visit of by Mareike Hasenpflug, consists in designing and validating slice samplers for distributions on manifolds. It is mildly connected to some current work on MCMC algorithms on manifolds through coupling techniques by [my friends & coauthors] Elena Bortolado, Pierre Jacob, and Robin Ryder (who escape temporarily the manifold at each step). As in Neal (2003), uniform draws from the (super)level sets are replaced there with one-step Markov moves within the level set, that is, slice sampler moves. The slice sampler actually generalises Neal’s (2003) stepping-out and shrinkage steps rather closely. Based on the standard notion of the Riemannian measure induced by the very structure of the manifold, the model therein assumes that the simulation target is available as a closed-form if unormalised density p(x) against that measure, meaning that problems where the distribution is a push-forward one induced by a mapping onto the manifold are not necessary manageable. The slide sampler is decomposed into choosing (1-dimensional) geodesics defined by the manifold (and generalising great circles), uniformly, and then sampling by this one-dimensional slice sampling over the geodesic, under the level set constraint. Meaning that those geodesics must be manageable enough. (Note that the concept of stepping-out does not mean that the chain ever escapes from the manifold.) Demonstrating the validity and reversibility proves a challenging task.
