  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.

Methodology;Probability;Machine Learning