Statistics seminar

Abstract: We derive error bounds for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion on a compact Riemannian manifold. Two estimators of linear functionals of invariant measure based on the discretized Markov process are considered: a time-averaging estimator and an ensemble-averaging estimator. Imposing no restrictions beyond a nominal level of smoothness on potential function, first-order error bounds, in discretization step size, on the bias and variances of both estimators are derived. We will also discuss conditions for extending analysis to the case of non-compact manifolds and different variants of the algorithm. We will present numerical illustrations with distributions on the manifolds of positive and negative curvature which verify the derived bounds.

Joint work with Karthik Bharath (University of Nottingham), Alexander Lewis (University of Gottingen) and Michael Tretyakov (University of Nottingham)