Transport map accelerated Markov chain Monte Carlo

TL;DR

This paper presents a novel MCMC acceleration method using optimal transport maps to transform proposals, resulting in significantly faster sampling from complex distributions with minimal target distribution information.

Contribution

It introduces an adaptive transport map framework that improves MCMC efficiency by transforming proposals into more effective non-Gaussian distributions, with a convex optimization approach that requires only sample-based target information.

Findings

Achieves order-of-magnitude speedups over standard MCMC methods.

Uses convex optimization with no gradient information from the target.

Enables parallel and adaptive updates for large sample sizes.

Abstract

We introduce a new framework for efficient sampling from complex probability distributions, using a combination of optimal transport maps and the Metropolis-Hastings rule. The core idea is to use continuous transportation to transform typical Metropolis proposal mechanisms (e.g., random walks, Langevin methods) into non-Gaussian proposal distributions that can more effectively explore the target density. Our approach adaptively constructs a lower triangular transport map-an approximation of the Knothe-Rosenblatt rearrangement-using information from previous MCMC states, via the solution of an optimization problem. This optimization problem is convex regardless of the form of the target distribution. It is solved efficiently using a Newton method that requires no gradient information from the target probability distribution; the target distribution is instead represented via samples. Sequential updates enable efficient and parallelizable adaptation of the map even for large numbers of samples. We show that this approach uses inexact or truncated maps to produce an adaptive MCMC algorithm that is ergodic for the exact target distribution. Numerical demonstrations on a range of parameter inference problems show order-of-magnitude speedups over standard MCMC techniques, measured by the number of effectively independent samples produced per target density evaluation and per unit of wallclock time.