Langevin diffusions and the Metropolis-adjusted Langevin algorithm

TL;DR

This paper clarifies Langevin diffusions on Riemannian manifolds and introduces a new position-dependent Metropolis-adjusted Langevin algorithm (MALA) that improves sampling efficiency by ensuring the correct invariant density.

Contribution

It proposes a novel position-dependent MALA based on a Langevin diffusion in Euclidean space with the desired invariant density, clarifying its relation to existing methods.

Findings

The new MALA demonstrates increased efficiency in simulations.

The paper clarifies the measure and invariant density for Langevin diffusions on manifolds.

The proposed diffusion differs from previous models in general, though they are equivalent in some cases.

Abstract

We provide a clarification of the description of Langevin diffusions on Riemannian manifolds and of the measure underlying the invariant density. As a result we propose a new position-dependent Metropolis-adjusted Langevin algorithm (MALA) based upon a Langevin diffusion in $\mathbb{R}^d$ which has the required invariant density with respect to Lebesgue measure. We show that our diffusion and the diffusion upon which a previously-proposed position-dependent MALA is based are equivalent in some cases but are distinct in general. A simulation study illustrates the gain in efficiency provided by the new position-dependent MALA.