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：Langevin动态被广泛用于高维、非Gaussian分布的取样,其密度已知到正常化常数。特别是,人们对未经调整的Langevin算法(ULA)非常感兴趣,该算法将Langevin动态直接分离,以估计对目标分布的预期值。我们研究了运输图的使用,该图将目标分布大致正常化,作为Langevin动态的先决条件并加速其趋同。特别是,我们表明,在连续的时间里,当运输图应用于Langevin动态时,结果是一个具有运输图所定义指标的Rangevin多元动态(RMLDD)。这一连接提出了更系统化的学习度量法,还产生了地图所描述的RMLD的替代离异性,我们对此进行了研究。此外,我们表明,在某些条件下,当运输图与ULA结合使用时,我们可以改进在$-Wasserstein距离上产出过程汇合的几何率。 数字结果可以补充我们的理论主张。