Resonances in reflective Hamiltonian Monte Carlo

• URL: http://arxiv.org/abs/2504.12374v1
• Date: Wed, 16 Apr 2025 18:00:00 GMT
• Title: Resonances in reflective Hamiltonian Monte Carlo
• Abstract summary: Reflective Hamiltonian Monte Carlo with inexact reflections exhibits slow mixing in high dimensions.<n>We show that the collective motion transitions between fluid-like and discretisation-dominated behaviour.<n>Low-dimensional toy models of the dynamics are constructed which reproduce the dominant features of the high-dimensional problem.
• Score: 1.4999444543328293
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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