使用 Langevin 扩散对流形进行采样和估计

Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion with invariant measure dμϕ∝e−ϕdvolg on a compact Riemannian manifold. Two estimators of linear functionals of μϕ 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 ϕ, 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 μϕ 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.

使用紧致黎曼流形上的不变测度 dμϕ∝e−ϕdvolg 对本质上定义的朗之万扩散进行离散化，得出采样和估计的误差界限。考虑了两种基于离散马尔可夫过程的 μϕ 线性泛函估计器：基于单个轨迹的时间平均估计器和基于多个独立轨迹的集合平均估计器。对 ϕ 不施加超出名义平滑度水平的任何限制，在离散化步长中，导出两个估计量的偏差和方差的一阶误差界限。误差阶数与欧几里得空间和平坦空间中的最优速率相匹配，并导致离散马尔可夫过程的不变测度 μϕ 和平稳测度之间的距离存在一阶界限。证明技术的通用性利用了两个偏微分方程和对应于朗之万扩散的算子半群之间的联系，使它们适合于研究与朗之万扩散相关的更通用的采样算法。讨论了将分析扩展到非紧流形情况的条件。正曲率和负曲率流形上的对数凹分布和其他分布的数值说明阐明了导出的边界并证明了采样算法的实用性。

https://arxiv.org/abs/2312.14882