Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion with invariant measure $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 variances 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. 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：针对紧致黎曼流形上具不变测度$d\mu_\phi \propto e^{-\phi} \mathrm{dvol}_g$的内蕴定义朗之万扩散离散化过程，推导了采样与估计的误差界。考虑两种基于离散化马尔可夫过程的$\mu_\phi$线性泛函估计量：基于单条轨迹的时间平均估计量，以及基于多条独立轨迹的系综平均估计量。在不对$\phi$施加超越标称光滑度水平的限制条件下，推导了两种估计量偏差与方差关于离散化步长的一阶误差界。误差阶次与欧氏空间及平坦空间中的最优速率一致，并由此得到不变测度$\mu_\phi$与离散化马尔可夫过程平稳测度之间距离的一阶界。证明技术具有普适性——通过利用两个偏微分方程与朗之万扩散对应算子半群之间的关联——使其适用于研究更广泛的与朗之万扩散相关的采样算法族。讨论了将分析扩展至非紧流形情形的条件。通过正负曲率流形上对数凹分布及其他分布的数值算例，阐释了所导出的误差界并展示了采样算法的实际效用。