| Summary: | We introduce a new framework for efficient sampling from complex probability distributions, using a combination of transport maps and the Metropolis-Hastings rule. The core idea is to use deterministic couplings 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 Markov chain Monte Carlo (MCMC) states, via the solution of an optimization problem. This optimization problem is convex regardless of the form of the target distribution and can be solved efficiently without 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.
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