Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion with invariant measure $\text{d}\mu_\phi \propto e^{-\phi} \mathrm{dvol}_g $ on a compact Riemannian manifold. Two estimators of linear functionals of $\mu_\phi $ based on the discretized Markov process are considered: a time-averaging estimator based on a single trajectory and an ensemble-averaging estimator based on multiple independent trajectories. Imposing no restrictions beyond a nominal level of smoothness on $\phi$, first-order error bounds, in discretization step size, on the bias and variance/mean-square error of both estimators are derived. The order of error matches the optimal rate in Euclidean and flat spaces, and leads to a first-order bound on distance between the invariant measure $\mu_\phi$ and a stationary measure of the discretized Markov process. This order is preserved even upon using retractions when exponential maps are unavailable in closed form, thus enhancing practicality of the proposed algorithms. Generality of the proof techniques, which exploit links between two partial differential equations and the semigroup of operators corresponding to the Langevin diffusion, renders them amenable for the study of a more general class of sampling algorithms related to the Langevin diffusion. Conditions for extending analysis to the case of non-compact manifolds are discussed. Numerical illustrations with distributions, log-concave and otherwise, on the manifolds of positive and negative curvature elucidate on the derived bounds and demonstrate practical utility of the sampling algorithm.

翻译：本文推导了在紧致黎曼流形上，利用具有不变测度 $\text{d}\mu_\phi \propto e^{-\phi} \mathrm{dvol}_g $ 的、内蕴定义的朗之万扩散的离散化进行采样与估计的误差界。考虑了两种基于离散化马尔可夫过程的线性泛函 $\mu_\phi $ 的估计器：基于单条轨迹的时间平均估计器，以及基于多条独立轨迹的系综平均估计器。在仅对 $\phi$ 施加名义光滑性要求、不引入其他限制的条件下，推导了关于离散化步长的一阶误差界，涵盖两种估计器的偏差与方差/均方误差。该误差阶与欧几里得空间及平坦空间中的最优速率相匹配，并导出了不变测度 $\mu_\phi$ 与离散化马尔可夫过程平稳测度之间距离的一阶界。即使在指数映射无法以闭合形式获得、需使用回撤映射的情况下，该误差阶依然得以保持，从而增强了所提算法的实用性。证明技术利用了朗之万扩散对应的两个偏微分方程及其算子半群之间的联系，其通用性使其适用于研究与朗之万扩散相关的更广泛采样算法类。文中讨论了将分析推广至非紧流形情形所需的条件。通过在正曲率与负曲率流形上对对数凹及非对数凹分布进行的数值示例，阐明了所得误差界，并验证了采样算法的实际效用。