Resonances in reflective Hamiltonian Monte Carlo
This addresses mixing problems in high-dimensional MCMC methods, which is incremental for computational statistics and physics simulations.
The paper tackled slow mixing in reflective Hamiltonian Monte Carlo with inexact reflections when targeting uniform distributions from Dirac delta initializations, showing that particles can spontaneously unmix due to resonances, with critical step size scaling as a power law in dimension.
In high dimensions, reflective Hamiltonian Monte Carlo with inexact reflections exhibits slow mixing when the particle ensemble is initialised from a Dirac delta distribution and the uniform distribution is targeted. By quantifying the instantaneous non-uniformity 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 fluid-like and discretisation-dominated behaviour, 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.