Monte Carlo on manifolds in high dimensions

TL;DR

This paper presents an efficient, scalable Markov Chain Monte Carlo algorithm for sampling probability distributions on high-dimensional manifolds defined by constraints, with applications in physics and materials science.

Contribution

It introduces a practical, easy-to-implement MCMC method that leverages linear algebra and sparsity to handle high-dimensional constrained sampling problems.

Findings

Algorithm scales to thousands of dimensions

Requires only one matrix factorization per proposal

Effective in complex, constrained physical systems

Abstract

We introduce an efficient numerical implementation of a Markov Chain Monte Carlo method to sample a probability distribution on a manifold (introduced theoretically in Zappa, Holmes-Cerfon, Goodman (2018)), where the manifold is defined by the level set of constraint functions, and the probability distribution may involve the pseudodeterminant of the Jacobian of the constraints, as arises in physical sampling problems. The algorithm is easy to implement and scales well to problems with thousands of dimensions and with complex sets of constraints provided their Jacobian retains sparsity. The algorithm uses direct linear algebra and requires a single matrix factorization per proposal point, which enhances its efficiency over previously proposed methods but becomes the computational bottleneck of the algorithm in high dimensions. We test the algorithm on several examples inspired by soft-matter physics and materials science to study its complexity and properties.