Abstract

Error bounds are derived for sampling and estimation using an Euler-type discretization of an intrinsically defined ergodic Langevin diffusion with an invariant measure on a compact Riemannian manifold. A time-averaging and ensemble-averaging estimators are considered.  The bias and mean-square error of both estimators are analyzed. The order of error matches the optimal rate in Euclidean and flat spaces and leads to a first-order bound on the distance between the invariant measure and a stationary measure corresponding to an intrinsic Euler scheme. This order is preserved even upon using retractions when exponential maps are unavailable in closed form. Numerical illustrations with distributions, log-concave and otherwise, on the manifolds of positive and negative curvature elucidate the derived bounds and demonstrate th e practical utility of the sampling algorithm. The talk is based on a joint work with Karthik Bharath (Nottingham, UK), Alexander Lewis (Göttingen, Germany), and Akash Sharma (Gothenburg, Sweden).