揭示高维反射哈密顿蒙特卡洛中混合缓慢的共振机制
Resonances in reflective Hamiltonian Monte Carlo
- 用Sinkhorn散度量化分布非均匀性，发现粒子集体运动在流体与离散化行为间切换
- 临界步长随维度呈幂律变化，在球体和立方体中均出现密度共振导致自发分层
- 构建低维简化模型复现高维核心现象，为调参提供理论依据
在高维情形下，当粒子集合从狄拉克δ分布初始化且目标为均匀分布时，使用不精确反射的反射哈密顿蒙特卡洛会出现混合缓慢问题。通过引入Sinkhorn散度量化分布的瞬时非均匀性，我们揭示了混合困难的主要机制。在球体和立方体中，发现粒子集体运动会在类流体行为与离散化主导行为之间转换，临界步长随维度呈幂律缩放。在两种状态下，粒子均可能自发分离，引发粒子密度的共振，导致前述问题。此外，构建了低维简化模型，能重现高维问题的主要特征。最后，将该动力学与精确哈密顿粒子流对比，并讨论了调参实践。
原文摘要 · Abstract (English)
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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