Resonances in reflective Hamiltonian Monte Carlo.

Abstract

In high dimensions, reflective Hamiltonian Monte Carlo with inexact reflections exhibits slow mixing when the particle ensemble is initialized from a Dirac δ distribution and the uniform distribution is targeted. By quantifying the instantaneous nonuniformity of the distribution with the Sinkhorn divergence, we elucidate the principal mechanisms underlying the mixing problems. In spheres and cubes, we show that the collective motion transitions between fluidlike and discretization-dominated behavior, with the critical step size scaling as a power law in the dimension. In both regimes, the particles can spontaneously unmix, leading to resonances in the particle density and the aforementioned problems. Additionally, low-dimensional toy models of the dynamics are constructed which reproduce the dominant features of the high-dimensional problem. Finally, the dynamics is contrasted with the exact Hamiltonian particle flow and tuning practices are discussed.