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. We also show that applying a transport map to an irreversibly-perturbed ULA results in a geometry-informed irreversible perturbation (GiIrr) of the original dynamics. These connections suggest more systematic ways of learning metrics and perturbations, and also yield alternative discretizations of the RMLD described by the map, which we study. Under appropriate conditions, these discretized processes can be endowed with non-asymptotic bounds describing convergence to the target distribution in 2-Wasserstein distance. Illustrative numerical results complement our theoretical claims.

翻译：朗之万动力学被广泛应用于对高维、非高斯分布的采样，这些分布的密度函数除归一化常数外已知。特别地，非调整朗之万算法（ULA）因其直接离散化朗之万动力学以估计目标分布期望的特性而备受关注。我们研究了利用传输映射近似归一化目标分布的方法，以预条件化并加速朗之万动力学的收敛。我们证明，在连续时间情形下，当传输映射作用于朗之万动力学时，结果将形成以传输映射定义度量的黎曼流形朗之万动力学（RMLD）。同时，将传输映射应用于不可逆扰动的ULA会引发原始动力学的几何感知不可逆扰动（GiIrr）。这些关联揭示了更系统地学习度量与扰动的方法，并生成了该映射所描述的RMLD的替代离散化方案。在适当条件下，这些离散化过程可赋予非渐近收敛界，描述其以2-瓦瑟斯坦距离收敛至目标分布的过程。数值算例进一步补充验证了我们的理论结论。