MCMC with Strings and Branes: The Suburban Algorithm (Extended Version)

Abstract

Motivated by the physics of strings and branes, we develop a class of Markov chain Monte Carlo (MCMC) algorithms involving extended objects. Starting from a collection of parallel Metropolis-Hastings (MH) samplers, we place them on an auxiliary grid, and couple them together via nearest neighbor interactions. This leads to a class of "suburban samplers" (i.e., spread out Metropolis). Coupling the samplers in this way modifies the mixing rate and speed of convergence for the Markov chain, and can in many cases allow a sampler to more easily overcome free energy barriers in a target distribution. We test these general theoretical considerations by performing several numerical experiments. For suburban samplers with a fluctuating grid topology, performance is strongly correlated with the average number of neighbors. Increasing the average number of neighbors above zero initially leads to an increase in performance, though there is a critical connectivity with effective dimension d_eff ~ 1, above which "groupthink" takes over, and the performance of the sampler declines.