Langevin dynamics are widely used in sampling high-dimensional, non-Gaussian distributions whose densities are known up to a normalizing constant. In particular, there is strong interest in unadjusted Langevin algorithms (ULA), which directly discretize Langevin dynamics to estimate expectations over the target distribution. We study the use of transport maps that approximately normalize a target distribution as a way to precondition and accelerate the convergence of Langevin dynamics. In particular, we show that in continuous time, when a transport map is applied to Langevin dynamics, the result is a Riemannian manifold Langevin dynamics (RMLD) with metric defined by the transport map. This connection suggests more systematic ways of learning metrics, and also yields alternative discretizations of the RMLD described by the map, which we study. Moreover, we show that under certain conditions, when the transport map is used in conjunction with ULA, we can improve the geometric rate of convergence of the output process in the $2$--Wasserstein distance. Illustrative numerical results complement our theoretical claims.

翻译：朗之万动力学被广泛用于采样高维、非高斯分布，这些分布的密度已知但归一化常数未知。特别是，非调节朗之万算法（ULA）因其直接离散化朗之万动力学以估计目标分布期望值而备受关注。我们研究使用传输映射近似归一化目标分布的方法，作为预处理和加速朗之万动力学收敛的一种手段。具体而言，我们证明在连续时间框架下，将传输映射应用于朗之万动力学时，结果将生成以传输映射定义度量的黎曼流形朗之万动力学（RMLD）。这一联系提示了更系统化的度量学习方法，并催生了由该映射描述的RMLD的替代离散化方案（我们对此进行了研究）。此外，我们证明在特定条件下，将传输映射与ULA结合使用时，可以改善输出过程在$2$--Wasserstein距离下的几何收敛速率。数值实例补充验证了我们的理论结论。