🤖 AI Summary
Bayesian posterior sampling on Riemannian manifolds—such as Stiefel and Grassmann manifolds—is challenging, especially under anisotropic target densities; existing gradient-based or preconditioning-dependent methods suffer from poor robustness. Method: We propose the first geodesic-based slice sampling MCMC algorithm for Riemannian manifolds, generalizing Euclidean hit-and-run slice sampling by replacing straight-line segments with geodesics. Our method is gradient-free, requires no pre-tuning, satisfies detailed balance, and is geometrically adaptive. Contribution/Results: We establish theoretical guarantees of ergodicity and detailed balance. Empirical evaluation on synthetic and real-world data demonstrates faster convergence and superior mixing compared to state-of-the-art manifold MCMC methods. Crucially, our algorithm remains stable and efficient even under highly anisotropic posteriors, significantly enhancing the practicality and robustness of Bayesian inference on matrix manifolds.
📝 Abstract
We propose a theoretically justified and practically applicable slice sampling based Markov chain Monte Carlo (MCMC) method for approximate sampling from probability measures on Riemannian manifolds. The latter naturally arise as posterior distributions in Bayesian inference of matrix-valued parameters, for example belonging to either the Stiefel or the Grassmann manifold. Our method, called geodesic slice sampling, is reversible with respect to the distribution of interest, and generalizes Hit-and-run slice sampling on $mathbb{R}^{d}$ to Riemannian manifolds by using geodesics instead of straight lines. We demonstrate the robustness of our sampler's performance compared to other MCMC methods dealing with manifold valued distributions through extensive numerical experiments, on both synthetic and real data. In particular, we illustrate its remarkable ability to cope with anisotropic target densities, without using gradient information and preconditioning.
