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. 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：朗之万动力学广泛应用于采样高维、非高斯分布，这些分布的密度已知但归一化常数未知。特别地，非调整朗之万算法（ULA）备受关注，它直接离散化朗之万动力学以估计目标分布上的期望。我们研究了利用近似归一化目标分布的运输映射作为预条件手段加速朗之万动力学收敛的方法。我们证明，在连续时间框架下，当将运输映射应用于朗之万动力学时，会产生以该映射定义度量的黎曼流形朗之万动力学（RMLD）。这一关联揭示了更系统的度量学习方法，并衍生出由该映射描述的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，当运输映射与ULA联合使用时，能够提升输出过程在2-瓦瑟斯坦距离下的几何收敛速率。数值实验结果佐证了我们的理论断言。